Lecture Notes

Spectral Theory

Roland Schnaubelt

These lecture notes are based on my course from the summer semester 2015.I kept the numbering and the contents of the results presented in the lectures(except for minor corrections and improvements). Typically, the proofs andcalculations in the notes are a bit shorter than those given in the lecture.Moreover, the drawings and many additional, mostly oral remarks from thelectures are omitted here. On the other hand, I have added several lengthyproofs not shown in the lectures (mainly about Sobolev spaces). These areindicated in the text.

I like to thank Bernhard Konrad for his support in preparing the versionof these lecture notes from 2010 and Heiko Hoffmann and Martin Meyriesfor several corrections and comments.

Karlsruhe, 25. Mai 2020 Roland Schnaubelt

Contents

Chapter 1. General spectral theory 11.1. Closed operators 11.2. The spectrum 4

Chapter 2. Spectral theory of compact operators 142.1. Compact operators 142.2. The Fredholm alternative 172.3. The Dirichlet problem and boundary integrals 222.4. Closed operators with compact resolvent 262.5. Fredholm operators 28

Chapter 3. Sobolev spaces and weak derivatives 343.1. Basic properties 343.2. Density and embedding theorems 443.3. Traces 583.4. Differential operators in Lp spaces 63

Chapter 4. Selfadjoint operators 674.1. Basic properties 674.2. The spectral theorems 72

Chapter 5. Holomorphic functional calculi 815.1. The bounded case 815.2. Sectorial operators 86

Bibliography 94

ii

CHAPTER 1

General spectral theory

General notation. X ‰ t0u, Y ‰ t0u and Z ‰ t0u are Banach spacesover C with norms ‖¨‖ (or ‖¨‖X etc.). The space

BpX,Y q “ tT : X Ñ Yˇ

ˇT is linear and continuousu

is endowed with the operator norm ‖T‖ “ sup‖x‖ď1‖Tx‖, and we abbreviate

BpXq :“ BpX,Xq.Let DpAq be a linear subspace of X and A : DpAq Ñ Y be linear. Then

A, or pA,DpAqq, is called linear operator from X to Y (and on X if X “ Y )with domain DpAq. We denote by

NpAq “ tx P DpAqˇ

ˇAx “ 0u

and RpAq “ ty P Yˇ

ˇ D x P DpAq with y “ Axu

the kernel and range of A.

1.1. Closed operators

We recall one of the basic examples of an unbounded operator: Let X “

Cpr0, 1sq be endowed with the supremum norm and let Af “ f 1 with DpAq “C1pr0, 1sq. Then A is linear, but not bounded. Indeed, the functions un PDpAq given by unpxq “ p1

?nq sinpnxq for n P N satisfy ‖un‖8 Ñ 0 and

‖Aun‖8 ě |u1np0q| “?nÑ8 as nÑ8.

However, if fn P DpAq “ C1pr0, 1sq fulfill fn Ñ f and Afn “ f 1n Ñ g inCpr0, 1sq as n Ñ 8, then f P DpAq and Af “ g (see Analysis 1 & 2). Thisobservation leads us to the following basic definition.

Definition 1.1. Let A be a linear operator from X to Y . The operatorA is called closed if for all xn P DpAq, n P N, such that there exists x “limnÑ8 xn in X and y “ limnÑ8Axn in Y , we have x P DpAq and Ax “ y.

Hence, limnÑ8pAxnq “ AplimnÑ8 xnq if both pxnq and pAxnq convergeand A is closed.

Example 1.2. a) It is clear that every operator A P BpX,Y q is closed(with DpAq “ X). On X “ Cpr0, 1sq the operator Af “ f 1 with DpAq “C1pr0, 1sq is closed, as seen above.

b) Let X “ Cpr0, 1sq. The operator Af “ f 1 with

DpAq “ tf P C1pr0, 1sqˇ

ˇ fp0q “ 0u.

is closed in X. Indeed, let fn P DpAq and f, g P X be such that fn Ñ fand Afn “ f 1n Ñ g in X as n Ñ 8. Again by Analysis 1 & 2, the functionf belongs to C1pr0, 1sq and f 1 “ g. Since 0 “ fnp0q Ñ fp0q as n Ñ 8, we

1

1.1. Closed operators 2

obtain f P DpAq. This means that A is closed on X. In the same way wesee that A1f “ f 1 with

DpA1q “ tf P C1pr0, 1sq

ˇ

ˇ fp0q “ f 1p0q “ 0u

is closed in X. There are many more variants.

c) Let X “ Cpr0, 1sq and Af “ f 1 with

DpAq “ C1c pp0, 1sq “ tf P C

1pr0, 1sqˇ

ˇ supp f Ď p0, 1su,

where the support supp f of f is the closure of tt P r0, 1sˇ

ˇ fptq ‰ 0u in r0, 1s.This operator is not closed. In fact, consider the maps fn P DpAq given by

fnptq “

#

0, 0 ď t ď 1n,

pt´ 1nq2 , 1n ď t ď 1,

for every n P N. Then, fn Ñ f and f 1n Ñ f 1 in X as nÑ8, where fptq “ t2.However, supp f “ r0, 1s and so f R DpAq.

d) Let X “ LppRdq, 1 ď p ď 8, and m : Rd Ñ C be measurable. DefineAf “ mf with

DpAq “ tf P Xˇ

ˇmf P Xu.

This is the maximal domain. Then A is closed. Indeed, let fn Ñ f andAfn “ mfn Ñ g in X as n Ñ 8. Then there is a subsequence such thatfnj pxq Ñ fpxq and mpxqfnj pxq Ñ gpxq for a.e. x P Rd, as j Ñ 8. Hence,

mf “ g in LppRdq and we thus obtain f P DpAq and Af “ g.

e) Let X “ L1pr0, 1sq, Y “ C, and Af “ fp0q with DpAq “ Cpr0, 1sq.Then A is not closed from X to Y . In fact, look at fn P DpAq given by

fnptq “

#

1´ nt, 0 ď t ď 1n,

0, 1n ď t ď 1,

for n P N. Then ‖fn‖1 “ 1p2nqÑ 0 as nÑ8, but Afn “ fnp0q “ 1. ♦

Definition 1.3. Let A be a linear operator from X to Y . The graph ofA is given by

GpAq “ tpx,Axq P X ˆ Yˇ

ˇx P DpAqu.

The graph norm of A is defined by ‖x‖A “ ‖x‖X`‖Ax‖Y . We write rDpAqsif we equip DpAq with ‖¨‖A.

Of course, ‖¨‖A is equivalent to ‖¨‖X if A P BpX,Y q. We endow X ˆ Ywith the norm ‖px, yq‖XˆY “ ‖x‖X`‖y‖Y . Recall that a seqeunce in XˆYconverges if and only if its components in X and in Y converge.

Lemma 1.4. Every linear operator A from X to Y satisfies the followingassertions.

(a) GpAq Ď X ˆ Y is a linear subspace.(b) rDpAqs is a normed vector space and A P BprDpAqs, Y q.(c) A is closed if and only if GpAq is closed in X ˆ Y if and only if

rDpAqs is a Banach space.(d) Let A be injective and put DpA´1q :“ RpAq. Then, A is closed from

X to Y if and only if A´1 is closed from Y to X.

1.1. Closed operators 3

Proof. Parts (a) and (b) follow from the definitions, and assertion (c)implies (d) since

GpA´1q “ tpy,A´1yqˇ

ˇ y P RpAqu “ tpAx, xqˇ

ˇx P DpAqu

is closed in Y ˆX if and only if GpAq is closed in XˆY . We next show (c).The operator A is closed if and only if for all xn P DpAq, n P N, and

px, yq P X ˆ Y with pxn, Axnq Ñ px, yq in X ˆ Y as n Ñ 8, we havex P DpAq and Ax “ y; i.e., px, yq P GpAq. This property is equivalent tothe closedness of GpAq. Since ‖px,Axq‖XˆY “ ‖x‖X ` ‖Ax‖Y , a Cauchysequence or a converging sequence in GpAq corresponds to a Cauchy or aconverging sequence in rDpAqs, respectively. So rDpAqs is complete if andonly if pGpAq, ‖¨‖XˆY q is complete if and only if GpAq Ď XˆY is closed.

The next result is a variant of the open mapping theorem.

Theorem 1.5 (Closed Graph Theorem). Let X and Y be Banach spacesand A be a closed operator from X to Y . Then A is bounded (i.e., ‖Ax‖ ďc ‖x‖ for some c ě 0 and all x P DpAq) if and only if DpAq is closed in X.In particular, a closed operator with DpAq “ X belongs to BpX,Y q.

Proof. “ð”: Let DpAq be closed in X. Then DpAq is a Banach spacefor ‖¨‖X and ‖¨‖A. Since ‖x‖X ď ‖x‖A for all x P DpAq, a corollary to theopen mapping theorem (see e.g. Corollary 4.29 in [Sc2]) shows that there isa constant c ą 0 such that ‖Ax‖Y ď ‖x‖A ď c ‖x‖X for all x P DpAq.

“ñ”: Let A be bounded and let xn P DpAq converge to x P X with respectto ‖¨‖X . Then ‖Axn´Axm‖Y ď c ‖xn´xm‖X , and so the sequence pAxnqnis Cauchy in Y . There thus exists y :“ limnÑ8Axn in Y . The closednessof A shows that x P DpAq; i.e., DpAq is closed in X.

We next show that Theorem 1.5 is wrong without completeness and givean example of a non-closed, everywhere defined operator.

Remark 1.6. a) Let T be given by pTfqptq “ tfptq, t P R, on CcpRq withsupremum norm. This linear operator is everywhere defined, unbounded andclosed. In fact, take fn, f, g P CcpRq such that fnptq Ñ fptq and pTfnqptq “tfnptq Ñ gptq uniformly for t P R as nÑ8. Then gptq “ tfptq for all t P R;i.e., g “ Tf and T is closed. Further, pick ϕn P CcpRq with ϕn8 “ 1 andϕnpnq “ 1. Since Tϕn8 ě |Tϕnpnq| “ n, the operator T is unbounded.

b) Let X be an infinite dimensional Banach space and let B be an algebraicbasis of X, see Theorem III.5.1 in [La]. (Hence, for each x P X there areunique α1, . . . , αn P C, b1, . . . , bn P B and n P N such that x “ α1b1 ` ¨ ¨ ¨ `αnbn.) We may assume that ‖b‖ “ 1 for all b P B. Choose a countablesubset B0 “ tbk

ˇ

ˇ k P Nu of B and set

Tbk “ kbk for each bk P B0, and Tb “ 0 for each b P BzB0.

Then T can be extended to a linear operator on X which is unbounded,since ‖Tbk‖ “ k and ‖bk‖ “ 1. Thus T is not closed by Theorem 1.5. ♦

Proposition 1.7. Let A be closed from X to Y , T P BpX,Y q, and S PBpZ,Xq. Then the following operators are closed.

(a) B “ A` T with DpBq “ DpAq,(b) C “ AS with DpCq “ tz P Z

ˇ

ˇSz P DpAqu.

1.2. The spectrum 4

Proof. (a) Let xn P DpBq, n P N, and x P X, y P Y such that xn Ñ xin X and Bxn “ Axn`Txn Ñ y in Y as nÑ8. Since T is bounded, thereexists Tx “ limnÑ8 Txn and so Axn Ñ y´Tx as nÑ8. The closedness ofA then yields x P DpAq “ DpBq and Ax “ y ´ Tx; i.e., Bx “ Ax` Tx “ y.

(b) Let zn P DpCq, n P N, and z P Z, y P Y such that zn Ñ z in Z andASzn Ñ y in Y as nÑ8. By the boundedness of S, the vectors xn :“ Sznconverge to Sz. Since Axn Ñ y and A is closed, we obtain Sz P DpAq andASz “ y; i.e., z P DpCq and Cz “ y.

Corollary 1.8. Let A be linear on X and λ P C. Then the followingassertions hold.

(a) A is closed on X if and only if λI ´A is closed on X.(b) If λI ´A is bijective with pλI ´Aq´1 P BpXq, then A is closed.

Proof. Assertion (a) is a consequence of Proposition 1.7 since A “

´ppλI ´ Aq ´ λIq. For the second part, Lemma 1.4 shows that λI ´ A isclosed, and then assertion (a) yields (b).

The following examples show that closedness can be lost when taking sumsor products of closed operators. See the exercises for further related results.

Example 1.9. a) Let E “ CbpR2q and Ak “ Bk with

DpAkq “ tf P Eˇ

ˇ the partial derivative Bkf exists and belongs to Eu,

for k P t1, 2u. Set B “ B1 ` B2 on

DpBq :“ DpA1q XDpA2q “ C1b pR2q “ tf P C1pR2q

ˇ

ˇ f, B1f, B2f P Eu.

By an exercise, A1 and A2 are closed. However, B is not closed.Indeed, take φn P C

1b pRq converging uniformly to some φ P CbpRqzC1pRq.

Set fnpx, yq “ φnpx ´ yq and fpx, yq “ φpx ´ yq for px, yq P R2 and n P N.We then obtain f P E, fn P DpBq, ‖fn ´ f‖8 “ ‖φn ´ φ‖8 Ñ 0 andBfn “ φ1n ´ φ

1n “ 0 Ñ 0 as nÑ8, but f R DpBq.

b) Let X “ Cpr0, 1sq, Af “ f 1 with DpAq “ C1pr0, 1sq and m P Cpr0, 1sqsuch that m “ 0 on r0, 12s. Define T P BpXq by Tf “ mf for all f P X.Then the operator TA with DpTAq “ DpAq is not closed.

To see this, take functions fn P DpAq such that fn “ 1 on r12, 1s andfn Ñ f in X with f R C1pr0, 1sq. Then, TAfn “ mf 1n “ 0 converges to 0,but f R DpAq. ♦

1.2. The spectrum

Definition 1.10. Let A be a closed operator on X. The resolvent set ofA is given by

ρpAq “ tλ P Cˇ

ˇλI ´A : DpAq Ñ X is bijectiveu,

and its spectrum by

σpAq “ CzρpAq.We further define the point spectrum of A by

σppAq “ tλ P Cˇ

ˇ D v P DpAqzt0u with λv “ Avu Ď σpAq,

1.2. The spectrum 5

where we call λ P σppAq an eigenvalue of A and the corresponding v aneigenvector or eigenfunction of A. For λ P ρpAq the operator

Rpλ,Aq :“ pλI ´Aq´1 : X Ñ X

and the set tRpλ,Aqˇ

ˇλ P ρpAqu are called the resolvent.

Remark 1.11. a) Let A be closed on X and λ P ρpAq. The resolventRpλ,Aq has the range DpAq. Corollary 1.8 and Lemma 1.4 further showthat Rpλ,Aq is closed and thus it belongs to BpXq by Theorem 1.5.

b) Let A be a linear operator such that λI´A : DpAq Ñ X has a boundedinverse for some λ P C. Then A is closed by Corollary 1.8. In this case, theclosedness assumption in Definition 1.10 is redundant. ♦

We set eλptq “ eλt for λ P C, t P J , and any interval J Ď R.

Example 1.12. a) Let X “ Cd and T P BpXq. Then σpT q only consistsof the eigenvalues λ1, . . . , λm of T , where m ď d.

b) Let X “ Cpr0, 1sq and Au “ u1 with DpAq “ C1pr0, 1sq. Then σpAq “σppAq “ C. Indeed, eλ belongs to DpAq and Aeλ “ λeλ for each λ P C.

c) Let X “ Cpr0, 1sq and Au “ u1 with DpAq “ tu P C1pr0, 1sqˇ

ˇup0q “ 0u.Then A is closed by Example 1.2. Moreover, σpAq is empty. In fact, letλ P C and f P X. We then have u P DpAq and pλI ´ Aqu “ f if and onlyif u P C1pr0, 1sq, u1ptq “ λuptq ´ fptq, t P r0, 1s, and up0q “ 0, which isequivalent to

uptq “ ´

ż t

0eλpt´sqfpsq ds “: pRλqfptq,

for all 0 ď t ď 1. Hence, σpAq “ H and Rpλ,Aq “ Rλ. ♦

We see in Examples 1.21 and 1.25 that both for unbounded and boundedoperators the point spectrum can be empty though the spectrum is void,provided that dimX “ 8.

Let U Ď C be open. The derivative of f : U Ñ Y at λ P U is given by

f 1pλq “ limµÑλ

1

µ´ λpfpµq ´ fpλqq P Y,

if the limit exists in Y . In the next theorem we collect the most basicproperties of the spectrum and the resolvent of closed operators.

Theorem 1.13. Let A be a closed operator on X and let λ P ρpAq. Thenthe following assertions hold.

(a) ARpλ,Aq “ λRpλ,Aq´I, ARpλ,Aqx “ Rpλ,AqAx for all x P DpAq,and1

µ´ λpRpλ,Aq ´Rpµ,Aqq “ Rpλ,AqRpµ,Aq “ Rpµ,AqRpλ,Aq

if µ P ρpAqztλu. The formula in display is called the resolventequation.

(b) The spectrum σpAq is closed, where Bpλ, 1‖Rpλ,Aq‖q Ď ρpAq and

Rpµ,Aq “8ÿ

n“0

pλ´ µqnRpλ,Aqn`1 “: Rµ

1.2. The spectrum 6

if |λ´ µ| ă 1‖Rpλ,Aq‖ “: rλ. This series converges in BpX, rDpAqsq,absolutely and uniformly on Bpλ, δrλq for each δ P p0, 1q. Moreover,

‖Rpµ,Aq‖BpX,rDpAqsq ďcpλq

1´ δ

for all µ P Bpλ, δ‖Rpλ,Aq‖q and a constant cpλq given by (1.1).(c) The function ρpAq Ñ BpX, rDpAqsq; λ ÞÑ Rpλ,Aq, is infinitely often

differentiable with`

ddλ

˘nRpλ,Aq “ p´1qn n!Rpλ,Aqn`1 for every n P N.

(d) ‖Rpλ,Aq‖ ě 1dpλ,σpAqq .

Proof. (a) The first assertions follow from

x “ pλI ´AqRpλ,Aqx “ Rpλ,AqpλI ´Aqx,

where x P X in the first equality and x P DpAq in the second one. Forµ P ρpAq, we further have

pλRpλ,Aq ´ARpλ,AqqRpµ,Aq “ Rpµ,Aq,

Rpλ,AqpµRpµ,Aq ´ARpµ,Aqq “ Rpλ,Aq.

Subtracting these two equations and also interchanging λ and µ, we deducethe resolvent equation.

(b) Let |µ´ λ| ď δ‖Rpλ,Aq‖ for some δ P p0, 1q and x P X with ‖x‖ ď 1. Itfollows

‖pλ´ µqnRpλ,Aqn`1x‖A

ďδn

‖Rpλ,Aq‖n`

‖ARpλ,AqRpλ,Aqnx‖` ‖Rpλ,Aqn`1x‖˘

ď δn p‖λRpλ,Aq‖` 1` ‖Rpλ,Aq‖q “: δncpλq. (1.1)

Due to this inequality and e.g. Lemma 4.23 in [Sc2], the series Rµ convergesand can be estimated as asserted. Part (a) yields

pµI ´AqRpλ,Aq “ pµ´ λqRpλ,Aq ` I.

Using this fact and that A belongs to BprDpAqs, Xq, we infer

pµI ´AqRµ “ ´8ÿ

n“0

pλ´ µqn`1Rpλ,Aqn`1 `

8ÿ

n“0

pλ´ µqnRpλ,Aqn “ I,

and similarly RµpµI ´ Aqx “ x for all x P DpAq. Hence, µ P ρpAq andRµ “ Rpµ,Aq.

Assertion (c) is a consequence of the power series expansion, as in thescalar case. Part (b) also implies (d).

Proposition 1.14. Let Ω Ď Rd, m P CpΩq, X “ CbpΩq, and Af “ mfwith DpAq “ tf P X

ˇ

ˇmf P Xu. Then A is closed,

σpAq “ mpΩq,

and Rpλ,Aqf “ 1λ´mf for all λ P ρpAq and f P X.

For every closed (resp., non-empty and compact) subset S Ď C there is aclosed (resp., bounded) operator B on a Banach space with σpBq “ S.

1.2. The spectrum 7

Proof. The closedness of A can be shown as in Remark 1.6. Let λ RmpΩq and g P CbpΩq. The function f :“ 1

λ´mg then belongs to CbpΩq and

λf ´mf “ g so that mf “ λf ´ g P CbpΩq. As a result, f P DpAq and f isthe unique solution in DpAq of the equation λf ´Af “ g. This means thatλ P ρpAq and Rpλ,Aqg “ 1

λ´mg.

In the case that λ “ mpzq for some z P Ω, we obtain

ppλI ´Aqfqpzq “ λfpzq ´mpzqfpzq “ 0

for every f P DpAq. Consequently, λI ´A is not surjective and so λ P σpAq.

We now conclude that σpAq “ mpΩq since the spectrum is closed.The final assertion follows from Example 1.12c) if S “ H. Otherwise,

consider Ω “ S. Define A and X as above. Then, σpAq “ S and A isbounded if S is compact (where CbpSq “ CpSq).

A similar result is valid in Lp-spaces, see e.g. Example IX.2.6 in [Co2].

Example 1.15. Let X “ C0pR`q “ tf P CpR`qˇ

ˇ limtÑ8 fptq “ 0u withthe supremum norm and Af “ f 1 with

DpAq “ C10 pR`q “ tf P C1pR`q

ˇ

ˇ f, f 1 P Xu.

As in Example 1.2 one sees that A is closed. Moreover, σpAq “ tλ P

Cˇ

ˇ Reλ ď 0u “: C´ and σppAq “ tλ P Cˇ

ˇ Reλ ă 0u.Proof. First note that for λ P C with Reλ ă 0, the function eλ belongs

to DpAq and Aeλ “ λeλ. Hence,

tλ P Cˇ

ˇ Reλ ă 0u Ď σppAq Ď σpAq,

so that C´ Ď σpAq by the closedness of the spectrum.Next, let Reλ ą 0 and f P X. We then have u P DpAq and λu´ Au “ f

if and only if u P X X C1pR`q and u1ptq “ λuptq ´ fptq for all t ě 0. Thisequation is uniquely solved by

uptq “

ż 8

teλpt´sqfpsqds “: pRλfqptq, t ě 0.

We still have to show that Rλf P X. Let ε ą 0. Then there is a numbertε ě 0 such that |fpsq| ď ε for all s ě tε. We can now estimate

|Rλfptq| ďż 8

tepReλqpt´sq|fpsq|ds ď ε

ż 8

0e´Reλr dr “

ε

Reλ,

for all t ě tε, where we substituted r “ s´ t. As a result, Rλf P X and soλ P ρpAq with Rλ “ Rpλ,Aq. We have thus shown that σpAq “ C´.

Finally, assume there is a number λ P iR and a function v P C10 pR`q with

v1 “ λv. It follows vptq “ eλpt´sqvpsq and so |vptq| “ |vpsq| for all t ě s ě 0.Letting t Ñ 8, we infer that |vpsq| “ 0 for all s ě 0; i.e., v “ 0. Hence, Ahas no eigenvalues on iR. l

Complementing Theorem 1.13, we state additional properties of the spec-trum if the operator is bounded.

Theorem 1.16. Let T P BpXq. Then σpT q is a non-empty compact set.The spectral radius rpT q :“ maxt|λ|

ˇ

ˇλ P σpT qu is given by

rpT q “ limnÑ8

‖Tn‖1n “ infnPN

‖Tn‖1n ď ‖T‖,

1.2. The spectrum 8

and for λ P C with |λ| ą rpT q we have

Rpλ, T q “8ÿ

n“0

λ´n´1Tn “: Rλ.

Proof. 1) Since Tn`m ď Tn Tm for all n,m P N, by an elemen-tary lemma (see Lemma VI.1.4 in [We]) there exists the limit

limnÑ8

‖Tn‖1n “ infnPN

‖Tn‖1n “: r ď ‖T‖.

If |λ| ą r, then

lim supnÑ8

‖λ´nTn‖1n “1

|λ|limnÑ8

‖Tn‖1n “r

|λ|ă 1.

Using Lemma 4.23 in [Sc2], as in Analysis 1 one now checks that the seriesRλ converges in BpXq for |λ| ą r. Moreover,

pλI ´ T qRλ “8ÿ

n“0

λ´nTn ´8ÿ

n“0

λ´n´1Tn`1 “ I,

and similarly RλpλI ´ T q “ I. Hence, λ P ρpT q and Rλ “ Rpλ, T q. Dueto its closedness, the spectrum σpT q Ď Bp0, rq is compact. Therefore, rpT qexists as the maximum of a compact subset of R, and rpT q ď r.

2) Take Φ P BpXq˚ and define fΦpλq :“ ΦpRpλ, T qq for λ P D “

CzBp0, rpT qq. Note that fΦ : D Ñ C is complex differentiable and

fΦpλq “8ÿ

n“0

λ´n´1ΦpTnq “: Sλ if |λ| ą r.

By e.g. Theorem V.1.11 in [Co1], there are unique coeffcients am P C with

fΦpλq “8ÿ

m“´8

amλm for λ P D.

Hence, the series Sλ converges for all λ P D and so

@ λ P D, Φ P BpXq˚ : supnPN

|λ´n´1ΦpTnq| ă 8.

A corollary to the uniform boundedness principle (see e.g. Corollary 5.12 in[Sc2]) thus yields that

cpλq :“ supnPN

λ´n´1Tn ă 8

for each λ P D. As a result,

limnÑ8

‖Tn‖1n “ limnÑ8

|λ| p|λ| ‖λ´n´1Tn‖q1n ď |λ| limnÑ8

p|λ| cpλqq1n “ |λ|

for all |λ| ą rpT q. This means that r “ rpT q.3) Suppose that σpT q “ H. The functions fΦ from part 2) are then

holomorphic on C for every Φ P BpXq˚. Step 1) implies that

|fΦpλq| ď ‖Φ‖|λ|´18ÿ

n“0

‖T‖n

|λ|nď

2‖Φ‖|λ|

,

for all λ P C with |λ| ě 2‖T‖. Hence, fΦ is bounded and thus constant byLiouville’s theorem from complex analysis. The above estimate then shows

1.2. The spectrum 9

that ΦpRpλ, T qq “ 0 for all λ P C and Φ P BpX˚q. Employing the Hahn-Banach theorem (see e.g. Corollary 5.10 in [Sc2]), we obtain Rpλ, T q “ 0,which is impossible since Rpλ, T q is injective and X ‰ t0u.

Example 1.17. a) We define the Volterra operator V on X“Cpr0, 1sq by

V fptq “

ż t

0fpsq ds

for t P r0, 1s and f P X. Then V belongs to BpXq and

|V nfptq| ďż t

0

ż s1

0. . .

ż sn´1

0‖f‖8 dsn . . . ds1 ď

1

n!‖f‖8,

for all n P N, t P r0, 1s, and f P X. Hence, ‖V n‖ ď 1n!. For f “ 1 we obtain‖V n‖ ě ‖V n

1‖8 “ 1n! and so ‖V n‖ “ 1n!. Theorem 1.16 thus yields

rpV q “ limnÑ8

ˆ

1

n!

˙1n

“ 0 and σpV q “ t0u.

Observe that σppV q “ H since V f “ 0 implies that f “ pV fq1 “ 0. More-over, ‖V ‖ “ 1 ą rpV q “ 0.

b) Let left shift L given by Lx “ pxn`1q on X P tc0, `pˇ

ˇ 1 ď p ď 8u has

the spectrum σpLq “ Bp0, 1q. Moreover, σppLq “ Bp0, 1q if X ‰ `8 and

σpLq “ Bp0, 1q if X “ `8.Proof. The operator L P BpXq has norm 1 (see e.g. Example 2.9 in

[Sc2]), and so σpLq Ď Bp0, 1q. Clearly, Lp1, 0, ¨ ¨ ¨ q “ 0. For |λ| ă 1 thesequence v “ pλnqnPN belongs to X and satisfies Lv “ pλn`1qn “ λv so thatBp0, 1q Ď σppLq Ď σpLq. If |λ| “ 1 and Lx “ λx, then xn`1 “ λxn for alln P N. Hence, x “ pλn´1x1qn which belongs to X (for x1 ‰ 0) if and only ifX “ `8. The closedness of the spectrum now implies the assertions. l

For further investigations we need certain subdivisions of the spectrum.We note that in this context also other definitions are used in the literature.

Definition 1.18. Let A be a linear operator on X. Then we call

σappAq “ tλ P Cˇ

ˇ D xn P DpAq with ‖xn‖ “ 1 for all n P N and

λxn ´Axn Ñ 0 as nÑ8u

the approximate point spectrum of A and

σrpAq “ tλ P Cˇ

ˇ pλI ´AqDpAq is not dense in Xu

the residual spectrum of A.

One calls λ P σappAq an approximate eigenvalue and the correspondingxn approximate eigenvectors. If one has λ P C and xn P DpAq with ‖xn‖ ěδ ą 0 for all n P N and λxn´Axn Ñ 0 as nÑ8, then λ belongs to σappAqwith approximate eigenvectors xn

´1xn, since xn´1 ď 1δ.

Proposition 1.19. Let A be closed on X. The following assertions aretrue (with possibly non-disjoint unions).

(a) σappAq “ σppAq Y tλ P Cˇ

ˇ pλI ´AqDpAq is not closed in Xu.(b) σpAq “ σappAq Y σrpAq.(c) BσpAq Ď σappAq.

1.2. The spectrum 10

Proof. 1) Let λ R σappAq. Then there is a constant c ą 0 with ‖pλI ´Aqx‖ ě c‖x‖ for all x P DpAq. This lower estimate implies that λ R σppAq.Moreover, if yn “ λxn´Axn Ñ y in X as nÑ8 for some xn P DpAq, thenthe lower estimate shows that pxnq is Cauchy in X, and so xn tends to somex in X. Hence, Axn “ λxn ´ yn Ñ λx ´ y and the closedness of A yieldsx P DpAq and λx´Ax “ y. Consequently, pλI ´AqDpAq is closed.

Conversely, if pλI ´ AqDpAq is closed and λ R σppAq, then the inversepλI ´ Aq´1 exists and is closed on its closed domain pλI ´ AqDpAq. Theclosed graph theorem 1.5 then yields the boundedness of pλI ´Aq´1. Thus,

‖x‖ “ ‖pλI ´Aq´1pλI ´Aqx‖ ď C‖pλI ´Aqx‖for all x P DpAq and a constant C ą 0. This means that λ R σappAq. Wethus have shown assertion (a), which implies (b).

2) Let λ P BσpAq. Then there are points λn in ρpAq with λn Ñ λ asn Ñ 8. By Theorem 1.13(d), the norms ‖Rpλn, Aq‖ tend to 8 as n Ñ 8,and thus there exist yn P X such that ‖yn‖ “ 1 for all n P N and 0 ‰ an :“‖Rpλn, Aqyn‖ Ñ 8 as n Ñ 8. Set xn “

1anRpλn, Aqyn P DpAq. We then

have ‖xn‖ “ 1 for all n P N and λxn ´Axn “ pλ´ λnqxn `1anyn converges

to 0 as nÑ8. As a result, λ P σappAq.

In the next result we determine the spectra of certain operators which(formally) arise as functions fpAq of A, namely the resolvent of A wherefpµq “ pλ ´ µq´1 for λ P ρpAq and a affine transformation of A wherefpµq “ αµ` β for α, β P C.

Proposition 1.20. Let A be closed on X, λ P ρpAq, α P Czt0u and β P C.Then the following assertions hold.

(a) σpRpλ,Aqqzt0u “ pλ´ σpAqq´1 “ t 1λ´ν

ˇ

ˇ ν P σpAqu.

(b) σjpRpλ,Aqqzt0u “ pλ´ σjpAqq´1 for j P tp, ap, ru.

(c) If x is an eigenvector for the eigenvalue µ ‰ 0 of Rpλ,Aq, theny “ µRpλ,Aqx is an eigenvector for the eigenvalue ν “ λ ´ 1µ ofA. If y P DpAq is an eigenvector for the eigenvalue ν “ λ ´ 1µ ofA with µ P Czt0u, then x “ µ´1pλy ´Ayq is an eigenvector for theeigenvalue µ of Rpλ,Aq.

(d) rpRpλ,Aqq “ 1dpλ,σpAqq.(e) If A is unbounded, then 0 P σpRpλ,Aqq.(f) σpαA ` βIq “ ασpAq ` β and σjpαA ` βIq “ ασjpAq ` β for

j P tp, ap, ru.

Proof. For µ P Czt0u we have

µI ´Rpλ,Aq “´

`

λ´ 1µ

˘

I ´A¯

µRpλ,Aq. (1.2)

Observe that the operator µRpλ,Aq : X Ñ DpAq is bijective. Hence, thebijectivity of µI´Rpλ,Aq is equivalent to that of pλ´ 1

µqI´A. As a result,

µ belongs to ρpRpλ,Aqq if and only if λ ´ 1µ belongs to ρpAq if and only if

µ “ pλ´ νq´1 for some ν P ρpAq. We have thus shown (a).In the same way, one derives assertion (b) for j “ p, assertion (c) and

that µI´Rpλ,Aq and pλ´ 1µqI´A have the same range. Using also Propo-

sition 1.19, we then deduce (b) for j “ ap and j “ r.

1.2. The spectrum 11

Part (d) is a consequence of (a). In part (e), the inverse Rpλ,Aq´1 “

λI´A is unbounded so that 0 P σpRpλ,Aqq. Similar as (a) and (b), the lastassertion follows from the equality

λI ´ pαA` βIq “ α`

λ´βα I ´A

˘

.

Example 1.21. a) Let X “ LppRq, 1 ď p ď 8, and the translation T ptqbe given by pT ptqfqpsq “ fps ` tq for s P R, f P X, and t P R. ThenσpT ptqq “ BBp0, 1q for t ‰ 0.

Proof. Recall from Example 4.12 in [Sc2] that T ptq is an isometry onX with inverse pT ptqq´1 “ T p´tq for every t P R. By Theorem 1.16 we haveσpT ptqq Ď Bp0, 1q. Proposition 1.20 further yields σpT ptqq´1 “ σpT ptq´1q “

σpT p´tqq Ď Bp0, 1q so that σpT ptqq Ď BBp0, 1q for all t P R. Fix t ‰ 0. Forevery λ P iR, the function eλ belongs to CbpRq Ď L8pRq and

pT ptqeλqpsq “ eλps`tq “ eλteλpsq

for all s P R. Hence, σpT ptqq “ σppT ptqq “ BBp0, 1q for p “ 8.If p P r1,8q, we use eλ to construct approximate eigenfunctions. For

n P N set fn “ n´1p1r0,nseλ. We then have ‖fn‖p “ n´1p‖1r0,ns‖p “ 1 and

(see above)

‖T ptqfn ´ eλtfn‖p “ n´1p‖eλtp1r´t,n´ts ´ 1r0,nsq‖p “ n´

1p|2t|´1p Ñ 0,

as nÑ8. As a result, σpT ptqq “ BBp0, 1q if t ‰ 0. l

b) Let X “ C0pRq and Au “ u1 with DpAq “ C10 pRq :“ tf P C1pRq

ˇ

ˇf, f 1 PC0pRqu. Then σpAq “ iR and σppAq “ H.Proof. As in Example 1.15 one sees that λ P ρpAq if Reλ ‰ 0 and

Rpλ,Aqfptq “

ż 8

teλpt´sqfpsqds if Reλ ą 0,

Rpλ,Aqfptq “ ´

ż t

´8

eλpt´sqfpsqds if Reλ ă 0,

for all t P R and f P X. Let Reλ “ 0. Then λ is not an eigenvalue, cf.Example 1.15. Choose ϕn P C

1c pRq with ‖ϕ1n‖8 ď 1n and ‖ϕn‖8 “ 1, and

set un “ ϕneλ for all n P N. Then, ‖un‖8 “ 1, un P DpAq and Aun “ϕ1neλ ` ϕne1λ “ ϕ1neλ ` λun. Since ‖ϕ1neλ‖8 ď 1n, we obtain λ P σappRq. l

We now introduce the adjoint of a densely defined linear operator in orderto obtain a convenient description of the residual spectrum, for instance.

Definition 1.22. Let A be a linear operator from X to Y with densedomain. We define its adjoint A˚ from Y ˚ to X˚ by setting

DpA˚q “ ty˚ P Y ˚ˇ

ˇ D z˚ P X˚ @x P DpAq : xAx, y˚y “ xx, z˚yu,

A˚y˚ “ z˚.

Observe that for all x P DpAq and y˚ P DpA˚q we obtain

xAx, y˚y “ xx,A˚y˚y.

We note that the operator Af “ f 1 with DpAq “ tf P C1pr0, 1sqˇ

ˇ fp0q “ 0u

is not densely defined on X “ Cpr0, 1sq since DpAq “ tf P Xˇ

ˇ fp0q “ 0u.

1.2. The spectrum 12

Remark 1.23. Let A be linear from X to Y with DpAq “ X.a) Since DpAq is dense, there is at most one vector z˚ “ A˚y˚ as in

Definition 1.22, so that A˚ : DpA˚q Ñ X˚ is a map. It is clear that A˚ islinear. If A P BpX,Y q, then Definition 1.22 coincides with the definition ofA˚ in §5.4 of [Sc2], where DpA˚q “ Y ˚.

b) The operator A˚ is closed from Y ˚ to X˚.Proof. Let y˚n P DpA˚q, y˚ P Y ˚, and z˚ P X˚ such that y˚n Ñ y˚ in Y ˚

and z˚n :“ A˚y˚n Ñ z˚ in X˚ as nÑ8. Let x P DpAq. We derive

xx, z˚y “ limnÑ8

xx, z˚ny “ limnÑ8

xAx, y˚ny “ xAx, y˚y.

As a result, y˚ P DpA˚q and A˚y˚ “ z˚. l

c) If T P BpX,Y q, then the sum A ` T with DpA ` T q “ DpAq has theadjoint pA` T q˚ “ A˚ ` T ˚ with DppA` T q˚q “ DpA˚q.

Proof. Let x P DpAq and y˚ P Y ˚. We obtain

xpA` T qx, y˚y “ xAx, y˚y ` xx, T ˚y˚y.

Hence, y˚ P DppA` T q˚q if and only if y˚ P DpA˚q, and then pA` T q˚y˚ “A˚y˚ ` T ˚y˚. l

Theorem 1.24. Let A be a closed operator on X with dense domain.Then the following assertions hold.

(a) σrpAq “ σppA˚q.

(b) σpAq “ σpA˚q and Rpλ,Aq˚ “ Rpλ,A˚q for every λ P ρpAq.

Proof. (a) Due to a corollary of the Hahn-Banach theorem (see e.g.Corollary 5.13 in [Sc2]), the set pλI ´ AqDpAq is not dense in X if andonly if there is a vector y˚ P X˚zt0u such that xλx´ Ax, y˚y “ 0 for everyx P DpAq. This equation is equivalent to xAx, y˚y “ xx, λy˚y, which in turnmeans that y˚ P DpA˚qzt0u and A˚y˚ “ λy˚; i.e., λ P σppA

˚q.(b) Let λ P ρpAq. Take y˚ P DpA˚q and x P X. We compute

xx,Rpλ,Aq˚pλI ´A˚qy˚y “ xRpλ,Aqx, pλI ´A˚qy˚y

“ xpλI ´AqRpλ,Aqx, y˚y “ xx, y˚y,

using Definition 1.22 and that Rpλ,Aqx belongs to DpAq. It followsRpλ,Aq˚pλI ´A˚qy˚ “ y˚ so that λI ´A˚ is injective. Next, take x˚ P X˚.Set y˚ “ Rpλ,Aq˚x˚. For x P DpAq, we obtain

xpλI ´Aqx, y˚y “ xRpλ,AqpλI ´Aqx, x˚y “ xx, x˚y.

Hence, y˚ P DpA˚q and x˚ “ pλI ´ Aq˚y˚ “ pλI ´ A˚qy˚, where we useRemark 1.23(c). Consequently, the operator λI ´ A˚ is surjective, hencebijective with inverse Rpλ,A˚q “ Rpλ,Aq˚.

Conversely, let λ P ρpA˚q. Then λ does not belong to σppA˚q “ σrpAq,

see (a). Take x P DpAq. Due to a corollary of the Hahn-Banach theorem(see e.g. Corollary 5.10 in [Sc2]), there is a functional y˚ P X˚ such that‖y˚‖ “ 1 and xx, y˚y “ ‖x‖. As above, we calculate

x “ xx, y˚y “ xx, pλI ´A˚qRpλ,A˚qy˚y “ xpλI ´Aqx,Rpλ,A˚qy˚y

ď ‖Rpλ,A˚q‖ ‖λx´Ax‖;

i.e., λ does not belong to σappAq. Proposition 1.19 thus yields λ R σpAq.

1.2. The spectrum 13

Example 1.25. Let X P tc0, `pˇ

ˇ 1 ď p ď 8u. Let Rx “ p0, x1, x2, . . . q be

the right shift on X. Then σpRq “ Bp0, 1q, σppRq “ H, σrpRq “ Bp0, 1q forX “ `1, and σrpRq “ Bp0, 1q for X P tc0, `

pˇ

ˇ 1 ă p ă 8u.Proof. First, let X ‰ `8. From e.g. Example 5.44 of [Sc2] we know that

R˚ “ L, where the left shift L acts on `1 if X “ c0 and on `p1

otherwise.Since σpLq “ Bp0, 1q by Example 1.17, Theorem 1.24 yields σpRq “ σpR˚q “σpLq “ Bp0, 1q. Similarly, σrpRq “ σppLq “ Bp0, 1q if X “ c0 or X “ `p

with 1 ă p ă 8, and σrpRq “ σppLq “ Bp0, 1q if X “ `1.

If X “ `8, then R “ L˚ for L on `1 so that again σpRq “ Bp0, 1q.Clearly, Rx “ 0 yields x “ 0. If λx “ Rx “ p0, x1, x2, ¨ ¨ ¨ q and λ ‰ 0,

then 0 “ λx1 and so x1 “ 0. Iteratively one sees that x “ 0. Hence, R hasno eigenvalues. l

Let A be a linear operator from X to Y . Then a linear operator B fromX to Y is called A–bounded (or if relatively bounded with respect to A) ifDpAq Ď DpBq and B P BprDpAqs, Y q.

Remark 1.26. Let A and B be linear from X to Y with DpAq Ď DpBq.a) The operator B is A–bounded if and only if there are constants a, b ě 0

such thatBx ď a Ax ` b x (1.3)

for all x P DpAq. If X “ Y and there exists a point λ in ρpAq, then theA–boundedness of B is also equivalent to the boundedness of BRpλ,Aq. Forinstance, we then have (1.3) with a :“ BRpλ,Aq and b :“ |λ| a.

b) Let A be closed and let (1.3) be satisfied with a ă 1. Then A ` Bwith DpA`Bq “ DpAq is also closed, by an exercise. In view of a), the nextresult also requires that b in (1.3) is sufficiently small. ♦

Theorem 1.27. Let A be a closed operator on X and λ P ρpAq. Further,let B be linear on X with DpAq Ď DpBq. Assume that ‖BRpλ,Aq‖ ă 1.Then A`B with DpA`Bq “ DpAq is closed, λ P ρpA`Bq,

Rpλ,A`Bq “ Rpλ,Aq8ÿ

n“0

pBRpλ,Aqqn “ Rpλ,AqpI ´BRpλ,Aqq´1,

‖Rpλ,A`Bq‖ ď ‖Rpλ,Aq‖1´ ‖BRpλ,Aq‖

.

Proof. By e.g. Proposition 4.24 of [Sc2], the operator I ´ BRpλ,Aqhas the inverse

Sλ “8ÿ

n“0

pBRpλ,Aqqn

in BpXq. Hence, λI ´ A ´ B “ pI ´ BRpλ,AqqpλI ´ Aq : DpAq Ñ X isbijective with the bounded inverse Rpλ,AqSλ. Remark 1.11 thus yields theclosedness of A ` B on DpAq, and so λ P ρpA ` Bq. The asserted estimatealso follows from Proposition 4.24 of [Sc2].

The smallness condition in the above theorem is sharp in general: LetX “ C, a P C – BpCq, a ‰ 0, and b “ a. Then a is invertible, but a´ a “ 0is not. Here we have λ “ 0 and |bRp0, aq| “ |aa | “ 1.

CHAPTER 2

Spectral theory of compact operators

2.1. Compact operators

We first state a few facts from, e.g., Section 1.3 of [Sc2]. A non-emptysubset S Ď X is compact if each sequence in S has a subsequence thatconverges in S. Equivalently, S is compact if every open covering of S has afinite subcovering. We call S Ď X relatively compact if S is compact whichmeans that each sequence in S has a converging subsequence (with limit inS). Finally, S Ď X is relatively compact if and only if it is totally bounded ;i.e., for each ε ą 0 there are finitely many balls in X covering it, whereone may chose the centers in S. Recall that compact sets are bounded andclosed. The converse is true if and only if X has finite dimension.

Definition 2.1. A linear map T : X Ñ Y is called compact if TBp0, 1qis relatively compact in Y . The set of all compact operators is denoted byB0pX,Y q.

Remark 2.2. a) If T is compact, then TBp0, 1q is bounded and thus T isbounded; i.e., B0pX,Y q Ď BpX,Y q.

b) Let T : X Ñ Y be linear. Then the following assertions are equivalent.

piq T is compact.piiq T maps bounded sets of X into relatively compact sets of Y .piiiq For every bounded sequence pxnqn in X there exists a convergent sub-

sequence pTxnj qj in Y .

Proof. piq ñ piiq: If T is compact and B Ď X is bounded, then B Ď

Bp0, rq for some r ě 0 and TB Ď TBp0, rq “ rTBp0, 1q, which is compact.Hence, TB is compact. The implications piiq ñ piiiq ñ piq are clear. l

c) The space of operators of finite rank is defined by

B00pX,Y q “ tT P BpX,Y qˇ

ˇ dimTX ă 8u,

see Example 5.16 of [Sc2]. For T P B00pX,Y q, the set TBp0, 1q is relativelycompact, and hence B00pX,Y q Ď B0pX,Y q.

d) The identity I : X Ñ X is compact if and only if Bp0, 1q is compact ifand only if dimX ă 8. ♦

The next result says that B0pX,Y q is a closed two-sided ideal in BpX,Y q.

Proposition 2.3. The set B0pX,Y q is a closed linear subspace ofBpX,Y q. Let T P BpX,Y q and S P BpY, Zq. If one of the operators Tor S is compact, then ST is compact.

Proof. Let xk P X, k P N, satisfy supkPN‖xk‖ “: c ă 8.

14

2.1. Compact operators 15

1) Let T,R P B0pX,Y q be compact. If α P C, then αT is also compact.There further exists a converging subsequence pTxkj qj . Since pxkj qj is stillbounded, there is another converging subsequence pRxkjl ql. Hence, ppT `

Rqxkjl ql converges and so T `R P B0pX,Y q.

2) Let Tn P B0pX,Y q converge in BpX,Y q to some T P BpX,Y q as nÑ8.Since T1 is compact, there is a converging subsequence pT1xν1pjqqj . Because‖xν1pjq‖ ď c for all j, there exists a subsubsequence ν2 Ď ν1 such thatpT2xν2pjqqj converges. Note that pT1xν2pjqqj still converges. Iteratively, weobtain subsequences νl Ď νl´1 such that pTnxνlpjqqj converges for all l ě n.

Set um “ xνmpmq. Then pTnumqm converges as mÑ8 for each n P N. Letε ą 0. Fix an index N “ Nε P N such that ‖TN ´ T‖ ď ε. Let m ě k ě N .We then obtain

‖Tum ´ Tuk‖ ď ‖pT ´ TN qum‖` ‖TN pum ´ ukq‖` ‖pTN ´ T quk‖ď 2εc` ‖TN pum ´ ukq‖.

Therefore pTumq is a Cauchy sequence. So we have shown that T is compact.Hence, B0pX,Y q is closed.

3) Let S P B0pX,Y q. Since pTxkqk is bounded, there is a convergingsubsequence pSTxkj qj , so that ST is compact.

If T P B0pX,Y q, then there is a converging subsequence pTxkj qj and thusSTxkj converges. Again, ST is compact.

Remark 2.4. Strong limits of compact operators may fail to be compact.Consider, e.g., X “ `2 and Tnx “ px1, . . . , xn, 0, 0, . . . q for all x P X andn P N. Then Tn P B00pXq Ď B0pXq but Tnx Ñ x “ Ix as n Ñ 8 for everyx P X and I R B0pXq. ♦

Example 2.5. a) Let X P tCpr0, 1sq, Lppr0, 1sqˇ

ˇ 1 ď p ď 8u, Y “

Cpr0, 1sq, and k P Cpr0, 1s2q. Setting

Tfptq “

ż 1

0kpt, τqfpτq dτ,

for f P X and t P r0, 1s, we define the integral operator T : X Ñ Y for thekernel k. We claim that T P B0pX,Y q.Proof. By Analysis 2 or 3, the function Tf is continuous for all f P X

and T : X Ñ Y is linear. Since ‖Tf‖8 ď ‖k‖8‖f‖1 ď ‖k‖8‖f‖p (usingthat λpr0, 1sq “ 1), we obtain that T P BpX,Y q and thus TB is bounded inY . where B :“ BXp0, 1q. Moreover, for t, s P r0, 1s and f P B we have

|Tfptq ´ Tfpsq| ďż 1

0|kpt, τq ´ kps, τq| |fpτq|dτ

ď supτPr0,1s

|kpt, τq ´ kps, τq| ‖f‖1

ď supτPr0,1s

|kpt, τq ´ kps, τq| ‖f‖p.

The right hand side tends to 0 as |t´s|Ñ 0 uniformly in f P B, because k isuniformly continuous. Therefore TB is equicontinuous. The Arzela-Ascolitheorem (see e.g. Theorem 1.47 in [Sc2]) then implies that TB is relativelycompact. Hence, T P B0pX,Y q. l

2.1. Compact operators 16

b) Let X “ Cpr0, 1sq and V fptq “şt0 fpsq ds for t P p0, 1q and f P X. This

defines a bounded operator V on X with norm 1, see Example 1.17. Letf P Bp0, 1q. Then V f8 ď 1 and pV fq18 “ f8 ď 1. The Arzela-Ascolitheorem (see e.g. Corollary 1.48 in [Sc2]) then yields the compactness of V .

c) Let X “ L2pRq. For f P X, we define

Tfptq “

ż

Re´|t´s|fpsq ds, t P R.

Theorem 2.14 of [Sc2] yields that T P BpXq. We claim that T is notcompact.Proof. Take fn “ 1rn,n`1s. For n ą m in N, we compute fn2 “ 1 and

Tfn ´ Tfm22 ě

ż n`2

n`1

ˇ

ˇ

ˇ

ˇ

ż n`1

nes´t ds´

ż m`1

mes´t ds

ˇ

ˇ

ˇ

ˇ

2

dt

“

ż n`2

n`1e´2t

`

en`1 ´ en ´ em`1 ` em˘2

dt

ě 12pe

´2n´2 ´ e´2n´4qpen`1 ´ 2enq2

“ 12pe

´2 ´ e´4qpe´ 2q2 ą 0.

Hence, pTfnq has no converging subsequence. l

d) Let E “ L2pRdq and k P L2pR2dq. For f P E, we set

Tfpxq “

ż

Rdkpx, yqfpyq dy, x P Rd.

By Example 5.44 in [Sc2], this defines an operator T P BpEq. We claimthat T is compact.Proof. Proposition 4.13 of [Sc2] yields kn P CcpR2dq that converge to k

in E. Let Tn be the corresponding integral operators in BpEq. There is aclosed ball Bn Ď Rd such that supp kn Ď Bn ˆBn. We then have

Tnfpxq “

#

0, x P RdzBn,ş

Bnknpx, yqfpyq dy, x P Bn.

for f P E and n P N. Let Rnf “ f|Bn . Fix n P N and take a boundedsequence pfkq in E. Arguing as in (a), one finds a subsequence such thatpRnTnfkj qj has a limit g0 in CpBnq. Since Bn has finite measure, this se-

quence converges also in L2pBnq. Hence, Tnfkj tend in E to the 0–extension

g of g0 as j P 8, and so Tn is compact. Let f P BEp0, 1q. Holder’s inequalityin the inner integral yields

Tf ´ Tnf22 “

ż

Rd

ˇ

ˇ

ˇ

ˇ

ż

Rdpkpx, yq ´ knpx, yqqfpyq dy

ˇ

ˇ

ˇ

ˇ

2

dx

ď

ż

Rd

ż

Rd|kpx, yq ´ knpx, yq|

2 dy

ż

Rd|fpyq|2 dy dx

ď k ´ kn22

for all n P N. The operators Tn thus converge to T in BpEq so that T iscompact by Proposition 2.3. l

2.2. The Fredholm alternative 17

Summarizing, integral operators are usually compact if the base space iscompact or has finite measure, or if the kernel decays fast enough at infinity.

Proposition 2.6. Let T P B0pX,Y q and xnσÑ x in X as n Ñ 8 (i.e.,

xxn, x˚y Ñ xx, x˚y as n Ñ 8 for every x˚ P X˚). Then Txn converges to

Tx in Y as nÑ8.

Proof. Let y˚ P Y ˚. We have xTxn´ Tx, y˚y “ xxn´ x, T

˚y˚y Ñ 0 as

nÑ8 for each y˚ P Y ˚, hence TxnσÑ Tx as nÑ8.

Suppose that Txn does not converge to Tx. Then there were δ ą 0 and asubsequence such that

‖Txnj ´ Tx‖ ě δ ą 0

for all j P N. By compactness, there is a subsubsequence and a y P Y suchthat Txnjl Ñ y as l Ñ 8. On the other hand, Txnjl

σÑ Tx and Txnjl

σÑ y.

Since weak limits are unique, it follows that y “ Tx, which is impossible.

Theorem 2.7 (Schauder). An operator T P BpX,Y q is compact if andonly if its adjoint T ˚ P BpY ˚, X˚q is compact.

Proof. 1) Let T be compact and take y˚n P Y˚ with supnPN‖y˚n‖ “: c ă

8. The set K :“ TBXp0, 1q is a compact metric space for the restrictionof the norm of Y . Set fn :“ y˚n|K P CpKq for each n P N. Putting c1 :“maxyPK‖y‖ ă 8, we obtain

‖fn‖8 “ maxyPK

|xy, y˚ny| ď cc1

for every n P N. Moreover, pfnqnPN is equicontinuous since

|fnpyq ´ fnpzq| ď ‖y˚n‖ ‖y ´ z‖ ď c ‖y ´ z‖

for all n P N and y, z P K. The Arzela-Ascoli theorem then yields a subse-quence pfnj qj converging in CpKq. We thus deduce that

‖T ˚y˚nj ´ T˚y˚nl‖X˚ “ sup

‖x‖ď1|xx, T ˚py˚nj ´ y

˚nly| “ sup

‖x‖ď1|xTx, y˚nj ´ y

˚nly|

ď ‖fnj ´ fnl‖CpKqtends to 0 as j, l Ñ 8. This means that pT ˚y˚nj qj converges and so T ˚ iscompact.

2) Let T ˚ be compact. By step 1), the bi-adjoint T ˚˚ is compact. LetJX : X Ñ X˚˚ be the canonical isometric embedding. Proposition 5.54 in[Sc2] says that T ˚˚JX “ JY T , and hence JY T is compact by Proposition 2.3.If pxnq is bounded in X, we thus obtain a converging subsequence pJY Txnj qjwhich is Cauchy. Since JY is isometric, also pTxnj qj is Cauchy and thusconverges; i.e., T is compact.

2.2. The Fredholm alternative

We need some fact from functional analysis. For non-empty sets M Ď Xand N˚ Ď X˚ we define the annihilators

MK “ tx˚ P X˚ˇ

ˇ@y PM : xy, x˚y “ 0u,

KN˚ “ tx P Xˇ

ˇ@y˚ P N˚ : xx, y˚y “ 0u.

2.2. The Fredholm alternative 18

These sets are equal to X˚ or X if and only if M “ t0u or N˚ “ t0u,respectively, see e.g. Remark 5.21 in [Sc2]. For T P BpXq we have

RpT qK “ NpT ˚q, RpT q “ KNpT ˚q,

NpT q “ KRpT ˚q, RpT ˚q Ď NpT qK.(2.1)

In particular, RpT q is dense if and only if T ˚ is injective; and if RpT ˚q isdense, then T is injective. (See e.g. Proposition 5.46 in [Sc2].)

The following theorem extends fundamental results on matrices knownfrom linear algebra.

Theorem 2.8 (Riesz 1918, Schauder 1930). Let K P B0pXq and T “

I ´K. Then the following assertions hold.

(a) RpT q is closed.(b) dim NpT q ă 8 and codim RpT q :“ dimXRpT q ă 8.(c) T is bijective ô T is surjective ô T is injective ô T ˚ is bijective ô T ˚

is surjective ô T ˚ is injective.

More precisely, we have

dim NpT q “ codim RpT q “ dim NpT ˚q “ codim RpT ˚q. (2.2)

Corollary 2.9 (Fredholm alternative). Let L P B0pXq, λ P Czt0u, andy P X. Then one of the following alternatives holds.

A) λx “ Lx has only the trivial solution x “ 0.Then for every y P X there is a unique solution x P X of λx ´ Lx “ ygiven by x “ Rpλ, Lqy.

B) λx “ Lx has an n-dimensional solution space NpλI´Lq for some n P N.Then there are n linearly independent solutions x˚1 , . . . , x

˚n P X

˚ of λx˚ “L˚x˚. The equation λx ´ Lx “ y has a solution x P X if and only ifxy, x˚ky “ 0 for every k “ 1, . . . , n. Every z P X satisfying λz ´ Lz “ yis of the form z “ x` x0, where λx´ Lx “ y and x0 P NpλI ´ Lq.

Proof of Corollary 2.9. We set K “ 1λL P B0pXq and note that

λx ´ Lx “ y is equivalent to pI ´Kqx “ 1λy. By Theorem 2.8(b), we have

either dim NpI ´Kq “ 0 (case A) or dim NpI ´Kq “ n P N (case B).In the first case, I´K is bijective due to Theorem 2.8(c) which yields A).In the second case, Theorem 2.8(a) shows that RpI ´Kq is closed so that

RpI´Kq “ KNpI´K˚q by (2.1). We thus deduce the solvability conditionfrom case B) noting that dim NpI ´K˚q “ n due to (2.2). If x ´Kx “ yand z´Kz “ y, then z´x belongs to NpI´Kq, as required in case B).

Example 2.10. Let X “ Cpr0, 1sq and V fptq “şt0 fpsqds for t P r0, 1s

and f P X. Then

RpV q “ tg P C1pr0, 1sqˇ

ˇ gp0q “ 0u,

which is not closed in X. Moreover, NpV q “ t0u and V is compact byExamples 1.17 and 2.5, respectively. In this case, V f “ g can not be solvedfor all g P X. Summing up, the Fredholm alternative fails for λ “ 0. ♦

Proof of Theorem 2.8. 1) The space N :“ NpT q “ T´1pt0uq isclosed in X. For x P N we have Kx “ x P N , so that K leaves N in-variant and its restriction KN to N coincides with the identity on N . Onthe other hand, KN is still compact so that dimN ă 8 by Remark 2.2d).

2.2. The Fredholm alternative 19

2) Since dimN ă 8, there is a closed subspace C Ď X such that NXC “t0u and N ` C “ X; i.e., X “ N ‘ C, see e.g. Proposition 5.17 in [Sc2].

Let T : C Ñ RpT q be the restriction of T to C and endow C and RpT qwith the norm of X. (Observe that C is a Banach space, but we do notknow yet whether RpT q is a Banach space.)

If T x “ 0 for some x P C, then x P N and so x “ 0. Let y P RpT q. Thenthere is an x P X such that Tx “ y. We can write x “ x0 ` x1 with x0 P Nand x1 P C. Hence, T x1 “ Tx1 ` Tx0 “ y, and so T is bijective.

A corollary to the open mapping theorem (see e.g. Corollary 4.31 in [Sc2])

now yields that RpT q “ RpT q is closed if and only if T´1 : RpT q Ñ C

is bounded. Assume that T´1 was unbounded. Then there would existelements yn “ T xn of RpT q with xn P C such that yn Ñ 0 as n Ñ 8 and

‖xn‖ “ ‖T´1yn‖ ě δ for some δ ą 0 and all n P N. We set xn “ xn´1xn

and note that xn “ 1 for all n P N and that

yn :“ T xn “1

‖xn‖yn

still converges to 0 as nÑ8. Since K is compact, there exists a subsequencepxnj qj and a vector z P X such that Kxnj Ñ z as j Ñ 8. Consequently,xnj “ ynj ` Kxnj Ñ z as j Ñ 8 and so ‖z‖ “ 1. Since C is closed, zbelongs to C. On the other hand, z is contained in N because of

Tz “ z ´Kz “ limjÑ8

Kxnj ´K limjÑ8

xnj “ 0.

Hence, z P C XN “ t0u, which contradicts ‖z‖ “ 1. Assertion (a) has beenestablished.

3) Theorem 2.7 shows the compactness of K˚ so that dim NpI´K˚q ă 8

by step 1). Using (2.1) and Proposition 5.23 in [Sc2], we further obtain

NpT ˚q “ RpT qK – pXRpT qq˚.

Since NpT ˚q is finite-dimensional, linear algebra yields that

dim NpT ˚q “ dimpXRpT qq˚ “ dimXRpT q “ codim RpT q (2.3)

and so codim RpT q ă 8; i.e., assertion (b) is true. We next prove in twosteps the remaining equalities in (2.2) which then yield the firt part of (c).1

4) Claim A: There is a closed linear subspace N with dim N ă 8 and a

closed linear subspace R of X such that

X “ N ‘ R, T N Ď N , T R Ď R and T2 :“ T|R : RÑ R is bijective.

Suppose for a moment that Claim A is true. Setting T1 :“ T|N P BpNq,

we obtain the following properties.(i) dim NRpT1q “ dim NpT1q (by the dimension formula in Cn).

(ii) NpT q “ NpT1q. In fact, writing x “ x1 ` x2 for x P X, x1 P N and

x2 P R we deduce that Tx “ 0 if and only if T2x2 “ ´T1x1 P N X R “ t0u.As T2 is injective, the latter statement is equivalent to x2 “ 0 and T1x1 “ 0.Hence, x P NpT q if and only if x P N and T1x “ 0; i.e., NpT q “ NpT1q.

1In the lectures (2.2) was not shown. Instead of steps 4) and 5) below, only the (mucheasier) proof of the first part of (c) was given.

2.2. The Fredholm alternative 20

(iii) We define the map

Φ : NRpT1q Ñ XRpT q; x` RpT1q ÞÑ x` RpT q,

for x P N Ď X. Because of RpT1q Ď RpT q, the map Φ is well defined. Ofcourse, it is linear. We want to show that Φ is bijective, which leads to

dim NRpT1q “ dimXRpT q.

Proof of (iii). If Φpx ` RpT1qq “ 0 for some x P N , then there is a vector

y P X with x “ Ty. There are y1 P N and y2 P R with y “ y1 ` y2. Hence,T2y2 “ x´T1y1 belongs to RX N “ t0u. Since T2 is injective, we infer thaty2 “ 0. Thus, Φ is injective.

Take x P X. There are x1 P N and x2 P R “ TR with x “ x1 ` x2. Wenow conclude that x´ x1 “ x2 P RpT q and so

Φpx1 ` RpT1qq “ x1 ` RpT q “ x` RpT q.

Hence, Φ is bijective. ♦The properties (i)–(iii) yield

dim NpT q “ dim NpT1q “ dim NRpT1q “ dimXRpT q “ codim RpT q. (2.4)

Since also K˚ is compact by Theorem 2.7, we further obtain

dim NpT ˚q “ codim RpT ˚q. (2.5)

Combining (2.3)–(2.5), we deduce assertion (c) assuming Claim A.5) Proof of Claim A. We set Nk “ NpT kq and Rk “ RpT kq for k P N0.

Observe that

t0u “ N0 Ď N1 Ď N2 Ď . . . , X “ R0 Ě R1 Ě R2 Ě . . . ,

TNk Ď Nk´1 Ď Nk, and TRk “ Rk`1 Ď Rk (2.6)

for all k P N0. We further have

T k “ pI ´Kqk “ I ´kÿ

j“1

ˆ

k

j

˙

p´1qj`1Kj “: I ´ Ck,

where Ck is compact for each k P N due to Proposition 2.3. Assertions (a)and (b) now imply that

Nk, Rk are closed and dimNk ă 8 (2.7)

for every k P N. We need four more claims to establish Claim A.

Claim 1: There is a minimal n P N0 such that Nn “ Nn`j for all j P N0.Indeed, if it was true that Nj Ř Nj`1 for all j P N0, then Riesz’ Lemma

(see e.g. Lemma 1.43 in [Sc2]) would give xj P Nj with ‖xj‖ “ 1 anddpxj , Nj´1q ě 12 for every j P N0. (Here we use that Nj´1 is closed.) Takel ą k ě 0. Since Txl ` xk ´ Txk P Nl´1 by (2.6), we deduce that

‖Kxl ´Kxk‖ “ ‖xl ´ pTxl ` xk ´ Txkq‖ ě 12.

As a result, pKxkqk has no converging subsequence, which contradicts thecompactness of K. So there is a minimal n P N0 with Nn “ Nn`1. Letx P Nn`2. Then, Tx P Nn`1 “ Nn so that x P Nn`1. This means thatNn`1 “ Nn`2, and Claim 1 follows by induction.

Claim 2: There is a minimal m P N0 such that Rm “ Rm`j for all j P N0.

2.2. The Fredholm alternative 21

Indeed, if it was true that Rj`1 Ř Rj for all j P N0, then Riesz’ Lemma(see e.g. Lemma 1.44 in [Sc2]) would give xj P Rj with ‖xj‖ “ 1 anddpxj , Rj`1q ě 12 for every j P N0. (Here we use that Rj`1 is closed.) Takel ą k ě 0. Since Txk ` xl ´ Txl P Rk`1 by (2.6), we deduce that

‖Kxk ´Kxl‖ “ ‖xk ´ pTxk ` xl ´ Txlq‖ ě 12.

As a result, pKxkqk has no converging subsequence, which contradicts thecompactness of K. So there is a minimal m P N0 with Rm “ Rm`1. Lety P Rm`1. Then there is a vector x P X such that y “ Tm`1x “ TTmx andhence y P TRm “ TRm`1 “ Rm`2 by (2.6). This means thatRm`1 “ Rm`2,and Claim 2 follows by induction.

Claim 3: Nn XRn “ t0u and Nm `Rm “ X.Indeed, for the first part, let x P Nn X Rn. Then Tnx “ 0 and there is a

vector y P X such that Tny “ x. Hence, T 2ny “ 0 and so y P N2n “ Nn byClaim 1. Consequently, x “ Tny “ 0.

For the second part, let x P X. Claim 2 yields that Tmx P Rm “ R2m,and thus there exists a vector y P X with Tmx “ T 2my. Therefore, x “px´ Tmyq ` Tmy belongs to Nm `Rm.

Claim 4: n “ m.Indeed, suppose that n ą m. Due to Claim 1 and Claim 2, there is an

element x P NnzNm and we have Rn “ Rm. Claim 3 further gives y P Nm Ď

Nn and z P Rm “ Rn such that x “ y ` z. Therefore, z “ x ´ y P Nn sothat z “ 0 by Claim 3. As a result, x “ y P Nm, which is impossible.

Second suppose that n ă m. Due to Claim 1 and Claim 2, we have Nn “

Nm and there is an element x P RnzRm. Claim 3 further gives y P Nm “ Nn

and z P Rm Ď Rn such that x “ y ` z. Therefore, y “ x ´ z P Rn so thaty “ 0 by Claim 3. As a result, x “ z P Rm, which is impossible.

We can now finish the proof of Claim A, setting N :“ Nn and R “ Rn.By (2.7), the spaces Nn and Rn are closed and dimNn ă 8. From Claims

3 and 4 we then infer that X “ N ‘ R. Moreover, (2.6) and Claim 3 yield

TN Ď N and TR “ R. If Tx “ 0 for some x “ Tny P R and y P X, theny P Nn`1 “ Nn by Claim 1. Therefore, x “ 0 and T

|R is bijective.

We next reformulate the Riesz-Schauder theorem in terms of spectraltheory. Observe that the Voltera operator in Example 2.10 is compact andhas the spectrum σpV q “ t0u with σppV q “ H.

Theorem 2.11. Let dimX “ 8 and K P BpXq be compact. Then thefollowing assertions hold.

(a) σpKq “ t0u 9Y tλjˇ

ˇ j P Ju, where J P tH,N, t1, . . . , nuˇ

ˇn P Nu.(b) σpKqzt0u “ σppKqzt0u. For all λ P σpKqzt0u the range of λI ´ K is

closed and

dim NpλI ´Kq “ codim RpλI ´Kq ă 8.

(c) For all ε ą 0 the set σpKqzBp0, εq is finite. Hence, λj Ñ 0 as j Ñ8 ifJ “ N.

Proof. If 0 P ρpKq, then K would be invertible. Proposition 2.3 nowshows that I “ K´1K would be compact which contradicts dimX “ 8.

2.3. The Dirichlet problem and boundary integrals 22

Hence, 0 always belongs to σpKq. Observe that thus assertion (a) followsfrom (c) by taking ε “ 1n for n P N.

For λ P Czt0u we have λI´K “ λpI´ 1λKq. Since 1

λK P B0pXq, Theorem2.8 implies either λ P ρpKq or λ P σppKq with

dim NpλI´Kq“ dim NpI´ 1λKq “ codim RpI´ 1

λKq “ codim RpλI´Kqă8.

So we have established (b).To prove assertion (c), we suppose that for some ε0 ą 0 we have points

λn in σpKqzBp0, ε0q with λn ‰ λm for all n ‰ m in N and vectors xn inXzt0u with Kxn “ λnxn. In Linear Algebra it is shown that eigenvectorsto different eigenvalues are linearly independent. Hence, the subspaces

Xn :“ lintx1, . . . , xnu

satisfy Xn Ř Xn`1 for every n P N. Moreover, KXn Ď Xn and Xn is closedfor each n P N (since dimXn ă 8). Riesz’ lemma (see e.g. Lemma 1.44 in[Sc2]) gives vectors yn in Xn such that ‖yn‖ “ 1 and dpyn, Xn´1q ě 12 foreach n P N. There are αn,j P C with yn “ αn,1x1 ` ¨ ¨ ¨ ` αn,nxn, and hence

λnyn ´Kyn “nÿ

j“1

pλn ´ λjqαn,jxj “n´1ÿ

j“1

pλn ´ λjqαn,jxj

belongs to Xn´1. For n ą m, the vector λnyn´Kyn`Kym is thus containedin Xn´1 so that

‖Kyn ´Kym‖ “ |λn|›

›

›yn ´

1

λnpλnyn ´Kyn `Kymq

›

›

›ě

|λn|2ěε0

2.

This fact contradicts the compactness of K.

Example 2.12. The following bounded operators are not compact sincetheir spectra are not finite or a null sequence.

a) the left and right shifts L and R on c0 or `p, 1 ď p ď 8, see Exam-ples 1.17 and 1.25;

b) the translation T ptqf “ fp¨ ` tq for t P Rzt0u, on LppRq, 1 ď p ď 8,see Example 1.21;

c) multiplication operators Tf “ mf on X “ CbpSq for S Ď Rd and

a given function m P CbpSq if mpSq is not finite or a null sequence, seeProposition 1.14. ♦

2.3. The Dirichlet problem and boundary integrals

In this subsection, we discuss a principal application of Theorem 2.8 topartial differential equations. Here we work with real-valued functions forsimplicity. We observe that Theorem 2.8 is still true for real scalars by thesame proof.

Let D Ď R3 be open, bounded and connected with BD P C2 and outerunit normal ν at BD (see part 3) below). Let ϕ P CpBDq be given.

Claim. There is a unique solution u in C2pDqXCpDq :“ tu P CpDqˇ

ˇu|D P

C2pDqu of the Dirichlet problem

∆upxq “ 0, x P D,

upxq “ ϕpxq, x P BD.(2.8)

2.3. The Dirichlet problem and boundary integrals 23

1) Tools from partial differential equations and uniqueness.We first state the strong maximum principle for the Laplacian, see The-orem 2.2.4 in [Ev].

(MP) Let u P C2pDq X CpDq satisfy ∆u “ 0 on D. Then maxD u “maxBD u. If there is a point x0 P D such that upx0q “ maxD u,then u is constant.

Hence, if u, v P C2pDqXCpDq solve (2.8), then w “ u´v P C2pDqXCpDqsatisfies ∆w “ 0 on D and w “ 0 on BD. The maximum principle (MP)thus yields that maxD w “ 0. Similarly, the maximum of ´w is 0, so thatw “ 0. This means that the problem (2.8) has at most one solution.

Theorem 2.7 of [PW] implies the following version of Hopf’s lemma.

(HL) Let u P C2pDq X CpDq satisfy ∆u ě 0 on D. Assume that there isa point x0 P BD such that upx0q “ maxD u, Bνupx0q exists, and Bνuis continuous at x0. Then either u is constant or Bνupx0q ą 0.

2) The double layer potential. We want to reformulate (2.8) usingan integral operator. To this aim, we first consider the “Newton potential”on R3 given by γpxq “ 1

4π|x|2 for x P R3zt0u. It satisfies ∆γ “ 0 on R3zt0u.

We define

kpx, yq “B

Bνpyqγpx´ yq “ ´p∇γqpx´ yq ¨ νpyq “ px´ yq ¨ νpyq

4π|x´ y|32(2.9)

for all x P R3 and y P BD with x ‰ y, where the dot denotes the Euclideanscalar product in R3. One introduces the “double layer potential” by setting

Sgpxq “

ż

BDkpx, yqgpyqdσpyq (2.10)

for all x P R3zBD and g P CpBDq, where the surface integral is recalledbelow in part 3). Standard results from Analysis 2 or 3 then imply thatSg P C8pR3zBDq and ∆Sg “ 0 on R3zBD. For each ϕ P CpBDq, one thusobtains the solution u “ Sg|D of (2.8) if one can find a map g P CpBDqsatisfying

limxÑzxPD

Sgpxq “ ϕpzq for all z P BD. (2.11)

3) The surface integral. A compact boundary BΩ of an open subset

Ω Ď Rd belongs to Ck, k P N, if there are open subsets Uj and Vj of Rd and

Ck–diffeomorphisms Ψj : Vj Ñ Uj , j P t1, . . . ,mu, such that the functions

Ψj and Ψ´1j and their derivatives up to order k have continuous extensions

to BVj and BUj , respectively, BΩ Ď V1Y¨ ¨ ¨Y Vm, and Ψj maps Vj :“ VjXBΩ

onto Uj :“ Uj X pRd´1 ˆ t0uq. We set Fj “ Ψ´1j |Uj . Below we use these

notions for k “ 2 and D “ Ω. We also identify Uj with a subset of R2

writing t P R2 instead of pt, 0q P R3.We recall that the surface integral for a (Borel) measurable function h :

BD Ñ R is given byż

BDhpyqdσpyq “

mÿ

j“1

ż

Uj

ϕjpFjptqqhpFjptqqb

detF 1ptqTF 1ptqdt,

2.3. The Dirichlet problem and boundary integrals 24

if the right hand side exists. Here, 0 ď ϕj P C8c pR3q satisfy suppϕj Ď Vj and

řmj“1 ϕj “ 1. This definition does not depend on the choice of Ψj : Vj Ñ Uj

and ϕj . Moreover, σpBq “ş

BD 1B dσ defines a (finite) measure on the Borelsets of BD. In particular, the above integral has the usual properties ofintegrals. We mostly omit the index j P t1, . . . ,mu.

Recall that for y “ F ptq P V and t P U Ď R2, the tangent plane of BDat y is spanned by B1F ptq and B2F ptq, where t P U Ď R2. Taylor’s formula

applied to Ψ´1 P C2b pUq at pt, 0q yields that

x :“ Ψ´1ps, 0q “ y ` pΨ´1q1pt, 0q

ˆ

s´ t

0

˙

`Op|s´ t|22q

“ y ` F 1ptqps´ tq `Op|s´ t|22q.

for s P U . Using that νpyq is orthogonal to BjF ptq, we deduce that

px´ yq ¨ νpyq “ νpyqTF 1ptqps´ tq `Op|s´ t|22q “ Op|s´ t|22q. (2.12)

On the other hand, Ψ´1 and Ψ are globally Lipschitz so that

c |s´ t|2 ď |x´ y|2 ď C |s´ t|2 (2.13)

for all x “ F psq P V , y “ F ptq P V with s, t P U and some constants C, c ą 0.In the following we denote by c various, possibly differing constants.

4) Compactness of a version of S on BD. Using (2.9), (2.12) and(2.13), we obtain

|kpx, yq| “ |px´ yq ¨ νpyq|4π |x´ y|32

ďc

|x´ y|2ď

c

|s´ t|2(2.14)

for all x “ F psq and y “ F ptq in V with x ‰ y. As a result, the integrands

ϕpF ptqqkpF psq, F ptqqgpF ptqqb

detF 1ptqTF 1ptq

of Sg are bounded by a constant times |s ´ t|´12 g8 for all x “ F psq and

y “ F ptq in BD with x ‰ y. We next set kpx, xq “ 0 for x P BD and

knpx, yq “

#

kpx, yq, |x´ y|2 ą 1n,

n3p4πq´1px´ yq ¨ νpyq, |x´ y|2 ď 1n,

for n P N. By means of (2.12) and (2.13), we estimate

|knpx, yq| ď cn3|s´ t|22 ď c|s´ t|´12 (2.15)

if |x´ y|2 ď 1n because then |s´ t|2 ď cn.Since kn is continuous on BD ˆ BD, we can define an operator Tn P

BpCpBDqq by

Tngpxq “

ż

BDknpx, yqgpyqdσ

for x P BD and g P CpBDq. As in Example 2.5a), one shows that Tn iscompact thanks to the Arzela-Ascoli theorem. For g P CpBDq and x P BD,we set Dpx, nq “ D XBpx, 1

nq and calculateż

BD|pkpx, yq ´ knpx, yqqgpyq|dσpyq “

ż

Dpx,nq|kpx, yq ´ knpx, yq| |gpyq|dσpyq

2.3. The Dirichlet problem and boundary integrals 25

ď g8

ż

Dpx,nqp|kpx, yq| ` |knpx, yq|q dσpyq

ď c ‖g‖8mÿ

j“1

ż

UjXBps,cnq

dt

|s´ t|2

ď c ‖g‖8ż

Bp0, cnq

dv

|v|2

ď c ‖g‖8ż cn

0

rdr

rďc ‖g‖8n

, (2.16)

employing (2.13), (2.14), (2.15), and polar coordinates in R2.Hence, for each x P BD the function y ÞÑ kpx, yqgpyq is integrable for the

surface measure σ on BD. So we can define

Tgpxq :“

ż

BDkpx, yqgpyqdσpyq,

for x P BD and g P CpBDq. By (2.16), the functions Tng converge uniformlyon BD to Tg as nÑ8 so that Tg P CpBDq. Estimate (2.16) actually impliesthat the differences Tn ´ T belong to BpCpBDqq and converge to 0 in thisspace. Hence, T is contained in BpCpBDqq and it is compact by Proposition2.3 since all Tn are compact.

5) Facts from potential theory. In Theorems VIII and IX in Chap-ter VI of [Ke] it is shown that

limxÑzxPD

Sgpxq “ Tgpzq ´ 12gpzq and lim

xÑzxPR3zD

Sgpxq “ Tgpzq ` 12gpzq, (2.17)

for all z P BD and g P CpBDq. We set

vpxq “

#

Sgpxq, x P R3zD,

gpxq, x P BD.(2.18)

If v “ 0 on D, then there exists Bνvpyq “ 0 for y P BD due to Theorem X inChapter VI of [Ke].

6) Conclusion. Let ϕ P pBDq. In view of (2.11) and (2.17), the functionSg P C2pDq has an extension u P C2pDqXCpDq solving (2.8) provided thatg P CpBDq satisfies 1

2g ´ Tg “ ´ϕ.Since T is compact, thanks to the Fredholm alternative Corollary 2.9 it

remains to establish the injectivity of 12I ´ T . So let g0 P CpBDq satisfy

12g0 “ Tg0. By the previous paragraph, the extension of Sg0 to D thensolves (2.8) with ϕ “ 0. This problem has also the trivial solution u “ 0.The uniqueness of (2.8) thus yields Sg0 “ 0 on D.

Define v0 by (2.18) with g0 instead of g. Then Bνv0 “ 0 on BD due tothe result mentioned after (2.18). We are now looking for a contradictionwith Hopf’s lemma (HL), employing Sg0 on R3zD. Fix r0 ą 0 such thatD Ď Bp0, r0q. For r ě r0 ` 1, x P BBp0, rq and y P BD, from (2.10) and(2.9) we deduce that

|kpx, yq| ď c

|x´ y|22ď

c

pr ´ r0q2ď

c

r2,

2.4. Closed operators with compact resolvent 26

|Sg0pxq| ďż

BD

c

r2‖g0‖8dσ ď

c

r2.

Suppose that g0 ‰ 0. We can thus fix a radius r ě r0 ` 1 such that

‖g0‖8 ą maxxPBBprq

|Sg0pxq|.

In particular, v0 is not constant on BprqzD. Since ∆v0 “ 0 on BprqzD, thestrong maximum principle (MP) says that v0 does not attain its maximum

on BprqzD. Since the maximum exists on BprqzD, it must be attained ata point y0 P BD, and hence v0py0q ą v0pxq for all x P BprqzD. Noting that´νpy0q is the outer unit normal of BprqzD at y0, we infer from (HL) thatBνv0py0q ă 0 contradicting Bνv0 “ 0 on BD. As a result, 1

2I ´ T is injectiveand we have established the claim.

2.4. Closed operators with compact resolvent

We now extend the results of Section 2.2 to a class of closed operatorsintroduced in the next definition.

Definition 2.13. A closed operator A on X has a compact resolvent ifthere exists a λ P ρpAq such that Rpλ,Aq P BpXq is compact.

Remark 2.14. a) Let A have a compact resolvent Rpλ,Aq and µ P ρpAq.The resolvent equation then yields

Rpµ,Aq “ Rpλ,Aq ` pλ´ µqRpλ,AqRpµ,Aq,

so that Rpµ,Aq is also compact due to Proposition 2.3.

b) Let A be closed onX with λ P ρpAq. Recall that rDpAqs “ pDpAq, ‖¨‖Aq.The following assertions are equivalent.

(i) A has a compact resolvent.(ii) Each bounded sequence in rDpAqs has a subsequence that converges in

X.(iii) The inclusion map J : rDpAqs Ñ X is compact.

Proof. Let (i) hold. Take xn P DpAq with ‖xn‖A ď c for n P N. Setyn “ λxn ´ Axn. Then ‖yn‖ ď p|λ| ` 1qc for every n P N so that xn “Rpλ,Aqyn has a subsequence which converges in X by (i), and (ii) is true.The implication ‘(ii) ñ (iii)’ follows from Remark 2.2. Let (iii) be valid.Define Rλ P BpX, rDpAqsq by Rλx “ Rpλ,Aqx for x P X. The operatorRpλ,Aq “ JRλ : X Ñ X is then compact due to Proposition 2.3. l

c) If T P BpXq has a compact resolvent, then dimX ă 8. In fact, theidentity I “ pλI´T qRpλ, T q would be compact by Proposition 2.3, if Rpλ, T qwas compact for some λ P ρpT q.

d) Let A be closed and densely defined on X with λ P ρpAq. Then, A hasa compact resolvent if and only if A˚ has a compact resolvent. Indeed, firstnote that λ P ρpA˚q andRpλ,Aq˚ “ Rpλ,A˚q by Theorem 1.24. Theorem 2.7then yields that the compactness of Rpλ,Aq and Rpλ,Aq˚ “ Rpλ,A˚q areequivalent. ♦

2.4. Closed operators with compact resolvent 27

In the next example we use the Holder space

CαpSq “!

f P CbpSqˇ

ˇ rf sα “ supx‰y

|fpyq ´ fpxq|

|x´ y|αă 8

)

for α P p0, 1q and S Ď Rd which is a Banach space with norm fCα “f8 ` rf sα. For α “ 1 the corresponding space is denoted by C1´pSq. Wehave C1´pSq ãÑ CαpSq ãÑ CβpSq ãÑ CbpSq if 0 ă β ď α ă 1 where theembeddings are given by the inclusion map.

Example 2.15. a) Let U Ď Rd be open and bounded, E be a closedsubspace of CpUq, and A be closed on E with λ P ρpAq. Assume thatDpAq ãÑ CαpUq for some α P p0, 1q. A bounded sequence pfnq in rDpAqsis thus bounded in CαpUq. The theorem of Arzela-Ascoli (see e.g. Corol-lary 1.48 in [Sc2]) yields a subsequence that converges in CpUq, and hencein E. By Remark 2.14, A has a compact resolvent in E.

b) Let X “ Cpr0, 1sq, Au “ u1 and DpAq “ tu P C1pr0, 1sqˇ

ˇup0q “ 0u.The spectrum of A is empty by Example 1.12. So part a) implies that Ahas compact resolvent.

c) Let X “ Cpr0, 1sq, Au “ u1 and DpAq “ C1pr0, 1sq. The resolvent setof A is empty by Example 1.12. Therefore A has no (compact) resolventalthough rDpAqs is compactly embedded in X by a). ♦

Theorem 2.16. Let dimX “ 8 and A be a closed operator with compactresolvent. Then the following assertions are true

(a) σpAq is either empty or σpAq “ σppAq contains at most countablymany eigenvalues λj.

(b) If σpAq is infinite, then |λj |Ñ8 as j Ñ8.(c) For all λj P σpAq, the range of λjI´A is closed and dimNpλjI´Aq “

codim RpλjI ´Aq ă 8.

Proof. Fix µ P ρpAq. By Theorem 2.11, the spectrum σpRpµ,Aqqonly contains 0 and either no or finitely many eigenvalues µj or a nullse-quence of eigenvalues µj . Moreover, the range of µjI ´ Rpµ,Aq is closedand dimNpµjI ´Rpµ,Aqq “ codim RpµjI ´Rpµ,Aqq ă 8 for all j. Propo-sition 1.20 yields pµ ´ σpAqq´1 “ σpRpµ,Aqqzt0u and pµ ´ σppAqq

´1 “

σppRpµ,Aqqzt0u. These facts imply assertions (a) and (b), where λj “

µ´ µ´1j . Observe that

λjI ´A “ µ´1j pµjI ´Rpµ,AqqpµI ´Aq.

Hence, also (c) follows from the results of Theorem 2.11 stated above.

Example 2.17. a) Let X “ Cpr0, 1sq and Au “ u1 with DpAq “ tu PC1pr0, 1sq

ˇ

ˇup0q “ up1qu. Then A is closed, has a compact resolvent andσpAq “ σppAq “ 2πiZ.

Proof. The closedness is shown as in Example 1.2. Let f P X. Afunction u belongs to DpAq and satisfies u ´ Au “ f if and only if u PC1pr0, 1sq, up0q “ up1q and u1 “ u´ f . These properties are equivalent to

uptq “ cet ´

ż t

0et´sfpsqds “: Rcfptq, t P r0, 1s, and up0q “ up1q,

2.5. Fredholm operators 28

for some c “ cpfq, which means that

c “ Rcfp0q “ Rcfp1q “ ce´ e

ż 1

0e´sfpsq ds,

c “e

e´ 1

ż 1

0e´sfpsqds.

We thus derive 1 P ρpAq and

Rp1, Aqfptq “et`1

e´ 1

ż 1

0e´sfpsq ds´

ż t

0et´sfpsqds

for t P r0, 1s. Due to Example 2.15a), the operator A thus has a compactresolvent and hence σpAq “ σppAq by Theorem 2.16. Finally, λ P C belongsto σppAq if and only if there is a non-zero u P C1pr0, 1sq with up0q “ up1q

and u1 “ λu which is equivalent to u “ eλup0q and 1 “ eλp0q “ eλp1q “ eλ,;i.e., λ P 2πiZ. l

b) Let X “ Cpr0, 1sq and Au “ u2 with DpAq “ tu P C2pr0, 1sqˇ

ˇup0q “up1q “ 0u. Then A is closed, has a compact resolvent and σpAq “ σppAq “t´π2k2

ˇ

ˇ k P Nu.Proof. We first note that uptq “ sinpπktq is an eigenfunction for A and

the eigenvalue λ “ ´π2k2, where k P N. Conversely, if λ P σppAq, we havea map u P C2pr0, 1sq with up0q “ up1q “ 0 and u2 “ λu. There thus exista, b, µ P C with µ2 “ λ and u “ aeµ ` be´µ ‰ 0. The conditions up0q “ 0and up1q “ 0 then yield a` b “ 0 and aeµ ` be´µ “ 0, respectively. Hence,e2µ “ 1 and µ ‰ 0; i.e., µ “ iπk and λ “ ´π2k2 for some k P Zzt0u. As ina), it remains to find a point in ρpAq. To this aim, set

Rfptq “1

2

ż 1

0|t´ s| fpsqds “

1

2

ż t

0pt´ sqfpsqds`

1

2

ż 1

tps´ tqfpsq ds,

for t P r0, 1s and f P X. Clearly, R is a bounded operator on X and

d

dtRfptq “

1

2

ż t

0fpsqds´

1

2

ż 1

tfpsqds,

so that Rf P C2pr0, 1sq with pRfq2 “ f . Then the function

R0fptq “ Rfptq ´ p1´ tqRfp0q ´ tRfp1q, t P r0, 1s,

belongs to DpAq and AR0f “ f . Since A is injective, we have shown thatA has the inverse R0. l

We note that in the above simple examples one could compute the resol-vent for all λ P ρpAq, cf. Example 5.9. In this sense the power of Theo-rem 2.16 is not really needed here.

2.5. Fredholm operators

In this section we study a class of operators which satisfy most of theassertions of the Riesz–Schauder Theorems 2.8 or 2.16. This class turns outto be stable under compact perturbations, and it arises in many applicationsas we indicate in Example 2.26.

2.5. Fredholm operators 29

Definition 2.18. A map T P BpX,Y q is called a Fredholm operator ifa) its range RpT q is closed in Y ,b) dim NpT q ă 8.c) codim RpT q “ dimY RpT q ă 8.In this case the index of T is the integer indpT q “ dim NpT q´codim RpT q.

For a closed operator A in Y we use the above definition with X “ rDpAqs.One sometimes calls dim NpT q the nullity and codim RpT q the defect of T .

Remark 2.19. a) An invertible operator T P BpX,Y q is Fredholm.Loosely speaking, Fredholmity means ‘invertibility except for finite dimen-sional spaces’, cf. Proposition 2.20 and the exercises.

b) Theorem 2.8 says that the operator λI ´K is Fredholm with index 0if K P BpXq is compact and λ P Czt0u. The same is true for λI ´A if λ P Cand A is closed in X with compact resolvent, by Theorem 2.16.

c) Each integer can occur as an index of a Fredholm operator. For in-stance, let T “ Ln for the left shift L on `p, 1 ď p ď 8, and some n P N;i.e., Tx “ pxn`kqk. Because of RpT q “ `p and NpT q “ linte1, . . . , enu, theoperator T is Fredholm with index n. Moreover, S “ Rn has index ´nfor the right shift R on `p and n P N since Sx “ p0, ¨ ¨ ¨ , 0, x1, x2, ¨ ¨ ¨ q withn zeros, S is injective, and `pRpSq – linte1, . . . , enu (cf. Example 2.20 of[Sc2]).

d) If dimX “ 8 then the Fredholm operators are not a linear subspaceof BpXq, since e.g. the identity I is Fredholm, but I ´ I “ 0 not.

e) One can omit condition a) in Definition 2.18.Proof. [Kato (1958)] Let T P BpX,Y q satisfy codim RpT q “ n P N0. If

n “ 0, then RpT q “ Y and we are done. Hence, take n P N. The operatorQ : X Ñ XNpT q; Qx “ x ` NpT q “ x, is a surjective contraction with

kernel NpT q, see e.g. Proposition 2.19 in [Sc2]. By T px`NpT qq “ T x :“ Tx,

we define a bijective operator T P BpXNpT q,RpT qq such that T “ TQ, cf.Proposition A.1.3 of [Co2]. Further, there are y1, ¨ ¨ ¨ , yn P Y such thatY “ RpT q ` linty1, ¨ ¨ ¨ , ynu and RpT q X linty1, ¨ ¨ ¨ , ynu “ t0u. On theBanach space E “ pXNpT qq ˆ Cn we define the operator

S : E Ñ Y ; Spx, pλ1, ¨ ¨ ¨ , λnqq “ T x`nÿ

j“1

λjyj “ Tx`nÿ

j“1

λjyj .

It straightforward to check that S is linear, bounded and bijective. Theopen mapping theorem (see e.g. Theorem 4.28 in [Sc2]) then says that S isan isomorphism. Consequently, RpT q “ SppXNpT qq ˆ t0uq is closed. l

In context of part e) of the above remark, we stress that for each infinitedimensional Banach space X there are non-closed subspaces Z of X withcodimension 1. To see this, take a countable subset B0 “ tbk

ˇ

ˇ k P Nu of analgebraic basis B of X as in Remark 1.6. Set ϕpbkq “ k and ϕpbq “ 0 forb P BzB0. Then ϕ extends to an unbounded linear map ϕ : X Ñ C. Defineϕ as in the proof of Remark 2.19e). It is a linear bijection from XNpϕqto Rpϕq “ C; i.e., codim Npϕq “ 1. However, Z “ Npϕq is not closed byProposition III.5.3 of [Co2].

2.5. Fredholm operators 30

The analysis in this section is based on the properties of Fredholm oper-ators established in the next result. (See the exercises for a related charac-terization of Fredholm operators of any index.)

Proposition 2.20. Let T P BpX,Y q be a Fredholm operator withindpT q “ 0. Then there exists an invertible operator J P BpY,Xq and afinite rank operator K P B00pXq such that JT “ IX ´K.

Proof. By the assumptions, there are closed subspaces X1 of X and Y0

of Y such that X “ NpT q‘X1, Y “ Y0‘RpT q and dimY0 “ codim RpT q “dim NpT q. (Use e.g. Proposition 5.17 of [Sc2].) We set

T1 : X1 ˆ Y0 Ñ Y ; T1px1, y0q “ Tx1 ` y0.

Clearly, T1 is linear and continuous. If T1px1, y0q “ 0 for some px1, y0q inX1 ˆ Y0, then y0 “ ´Tx1 P RpT q so that y0 “ 0. Hence, x1 belongs toNpT q X X1; i.e., x1 “ 0 and T1 is injective. Let y P Y . Then y “ y0 ` y1

for some y0 P Y0 and y1 “ Tx1 with x1 P X. As a result, y “ y0 ` Tx1 “

T1px1, y0q and T1 is bijective. The inverse T´11 : Y Ñ X1 ˆ Y0 is then

bounded by the open mapping theorem, see e.g. Theorem 4.28 in [Sc2].Observe that T´1

1 y1 “ px1, 0q if Tx1 “ y1 for some x1 P X1.There exists an isomorphism S : Y0 Ñ NpT q since these spaces have the

same finite dimension. Define

S1 : X1 ˆ Y0 Ñ Y ; S1px1, y0q “ x1 ` Sy0.

As above, one checks that S1 is invertible. We now introduce the invertibleoperator J “ S1T

´11 : Y Ñ X and the map K :“ IX ´ JT P BpXq. For

x1 P X1 we compute

Kx1 “ x1 ´ S1T´11 Tx1 “ x1 ´ S1px1, 0q “ x1 ´ x1 “ 0.

Since X “ NpT q ‘ X1, we derive KX Ď K NpT q, and hence dim RpKq ďdim NpT q ă 8 as asserted.

We can now show an important perturbation result for Fredholmity andthe index. The quantitative smallness condition from Theorem 1.27 (oninvertibility) is replaced by compactness.

Theorem 2.21. Let T P BpX,Y q be Fredholm and K P BpX,Y q be com-pact. Then T `K P BpX,Y q is Fredholm with indpT `Kq “ indpT q.

Proof. Set n “ indpT q P Z.1) Let n “ 0. Proposition 2.20 yields an invertible operator J P BpY,Xq

and a map K1 P B00pXq such that JT “ IX ´ K1. The product JK iscompact by Proposition 2.3. We thus deduce that JpT `Kq “ IX ´ pK1 ´

JKq is Fredholm with index 0 from Theorem 2.8. The invertibility of Jeasily implies that also T `K is Fredholm with index 0.

2) Let n ą 0. Set Y “ Y ˆ Cn, and define

T : X Ñ Y ; T x “ pTx, 0q, K : X Ñ Y ; Kx “ pKx, 0q.

It is straightforward to check that RpT q is closed, NpT q “ NpT q, Y RpT q –

pY RpT qq ˆ t0u. In particular, codim RpT q “ codim RpT q ` n and so T is

Fredholm with index 0. Since K is still compact, by part 1) the operator

2.5. Fredholm operators 31

T `K is also Fredholm with index 0. Noting that pT `Kqx “ pTx`Kx, 0q,we infer that T `K is Fredholm with index n.

3) Let n ă 0. Set X “ X ˆ C|n|, and define

T : X Ñ Y ; T px, ξq “ Tx, K : X Ñ Y ; Kpx, ξq “ Kx.

Starting from dim NpT q “ dim NpT q`|n| and RpT q “ RpT q, one derives theassertion as in part 2).

To exploit the above result in spectral theory, we need another definition.

Definition 2.22. For T P BpXq we define the essential spectrum by

σesspT q “ tλ P Cˇ

ˇλI ´ T is not Fredholmu.

We also set

σ0esspT q “ tλ P C

ˇ

ˇλI ´ T is not Fredholm of index 0u.

For a closed operator A on X we define analogously

σesspAq “ tλ P Cˇ

ˇλI ´A : rDpAqs Ñ X is not Fredholmu,

σ0esspAq “ tλ P C

ˇ

ˇλI ´A : rDpAqs Ñ X is not Fredholm of index 0u.

Observe that σesspBq Ď σ0esspBq Ď σpBq and that λ P σpBqzσ0

esspBq is aneigenvalue with finite dimensional eigenspace since λI´B is then Fredholmwith dim NpBq “ codim RpBq ă 8. (Here B P BpXq or B is closed on X.)

There are various differing concepts of the essential spectrum in the liter-ature. Typically, they lead to the same essential spectral radius

resspT q “ supt|λ|ˇ

ˇλ P σesspT qu,

if T P BpXq. The next concept is used below in our perturbation resultTheorem 2.25 for σesspAq.

Definition 2.23. Let A and B be linear from X to Y with DpAq Ď DpBq.Then B is called A–compact (or, relatively compact with respect to A) ifB : rDpAqs Ñ Y is compact.

Note that an A–compact operator is automatically A–bounded. Relativecompactness is further discussed in the excercises.

Lemma 2.24. Let A be closed from X to Y and B be A–compact. ThenA ` B with DpA ` Bq “ DpAq is closed and B is relatively compact withrespect to A`B.

Proof. 1) We first check that B is pA`Bq–compact. Let pxnq Ď DpAqbe bounded for ¨ A`B. In particular, pxnq is bounded in X.

Suppose that αn :“ Axn tends to infinity as n Ñ 8. Set xn “ α´1n xn

for n P N. (We may assume that Axn ‰ 0 for all n.) Then xn Ñ 0 in X andpA`Bqxn “ α´1

n pA`Bqxn Ñ 0 in Y as nÑ8, whereas Axn “ 1 for alln. Since B is A–compact, there is a subsequence such that Bxnj convergesto some y in Y as j Ñ 8. Hence, Axnj tends to ´y. The closedness of Athen yields y “ 0, which is impossible since 1 “ Axnj Ñ y as j Ñ8.

We conclude that there exists a subsequence such that pAxnkqk is boundedin Y . Using again the A–compactness of B, we obtain another subsequencepBxnklql with a limit in Y ; i.e., B is pA`Bq–compact.

2.5. Fredholm operators 32

2) Let xn P DpAq tend to some x in X and pA ` Bqxn to some y in Yas n Ñ 8. By part 1), there is subsequence and a vector z P Y such thatBxnj Ñ z in Y as j Ñ 8. As a result, Axnj tends to y ´ z. Since A isclosed, we infer that x P DpAq “ DpA ` Bq and Ax “ y ´ z. This meansthat xnj Ñ x in rDpAqs, and hence Bxnj Ñ Bx “ z by continuity. It followspA`Bqx “ y and so A`B is closed.

Theorem 2.25. Let A be closed on X and B be A–compact. Then

σesspA`Bq “ σesspAq and σ0esspA`Bq “ σ0

esspAq.

Proof. We apply Theorem 2.21 to the operators λI ´A and ´B fromrDpAqs to X, where λ P σesspAq or λ P σ0

esspAq, and to the operators λI´A´B and B from rDpA`Bqs to X, where λ P σesspA`Bq or λ P σ0

esspA`Bq.In the second part we use that B is pA ` Bq–compact by Lemma 2.24.Consequently, λI ´A is Fredholm (with index 0) if and only if λI ´A´Bis Fredholm (with index 0), as asserted.

We apply the above result to a typical situation arising in partial differ-ential equations. However, we can only give a rough sketch.

Example 2.26. We study the asymptotic stability of stationary solutionsto the reaction-convection-diffusion equation

Btupt, xq “ dBxxupt, xq ` cBxupt, xq ` fpupt, xqq, t ě 0, x P R,up0, xq “ u0pxq, x P R,

(2.19)

for diffusion and convection constants d ą 0 and c P R, a given initialstate u0 P CubpRq “ tv P CbpRq

ˇ

ˇ v is uniformly continuousu, and a function

f P C2pCq with fpRq Ď R describing (auto-)reaction (where we mean realdifferentiability). We interpret upt, xq as the density of a species.

It is known that there is a maximal existence time t “ tpu0q P p0,8s and aunique solution u of (2.19) in the space Cpr0, tq, CbpRqqXC1pp0, tq, CbpRqqXCpp0, tq, C2

b pRqq.2 Moreover, if u0 ě 0 and fp0q ě 0, then u ě 0. (See e.g.

Proposition 7.3.1 in [Lu] or Theorem 3.8 in [Sc3].)Let u˚ P C

2b pR,Rq be a stationary solution of (2.19); i.e., upt, xq “ u˚pxq

solves (2.19) which is equivalent to

0 “ du2˚ ` cu1˚ ` fpu˚q on R.

One now asks whether such special solutions describe well the behavior of(2.19), at least locally near u˚. One possible answer is the principle oflinearized stability. Here, one proceeds similar as for ordinary differentialequations. Define in X “ CbpRq the maps A and F by

Av “ dv2 ` cv1 with DpAq “ C2b pRq, F pvq “ f ˝ v.

One can then check that F P C1pX,Xq with derivative F 1pvq P BpXq atv P X given by F 1pvqw “ f 1pvqw for w P X. (One defines differentiablity inBanach spaces analogously as in Rn.) We introduce the linearized operatorat u˚ by setting

A˚v “ Av ` F 1pu˚qv “ dv2 ` cv1 ` f 1pu˚qv with DpA˚q “ C2b pRq.

2We note that one needs the uniform continuity of u0 to obtain the stated continuityof u at t “ 0.

2.5. Fredholm operators 33

The principle of linearized stability now says the following. If spA˚q :“suptReλ

ˇ

ˇλ P σpAqu ă ´δ ă 0, then there are constants c, r ą 0 such that

@ u0 P BXpu˚, rqXCubpRq : tpu0q “ 8 and uptq´u˚8 ď ce´δtu0´u˚8

for all t ě 0, where u solves (2.19). (See e.g. Theorem 3.8 in [Sc3].) Wenote that such results fail for certain partial differential equations. Here itworks since (2.19) is of ‘parabolic type’.

Of course, one now has to compute the sign of the spectral bound spA˚q (ordifferent properties of σpA˚q for more refined versions of the above result).We sketch a partial answer for the important special case that u˚psq haslimits ξ˘ in R as sÑ ˘8. Then the limit operators

A˘ “ A` F 1pξ˘1q with DpA˘q “ C2b pRq

have constant coefficients which simplifies the computation of their spectralproperties. We now follow the survey article [Sa].

Let λ P C. We rewrite A˘u´ λu “ g as a first order system by

Lpλq

ˆ

v1

v2

˙

:“

ˆ

v11v12

˙

´

ˆ

0 11dpλ´ f

1pξ˘qq ´ cd

˙ˆ

v1

v2

˙

“

ˆ

01dg

˙

,

where pv1, v2q fl pu, u1q. We set

M˘pλq “

ˆ

0 11dpλ´ f

1pξ˘qq ´ cd

˙

.

We denote by Xu˘pλq the linear span of all (generalized) eigenvectors of

M˘pλq for eigenvalues µ with Reµ ą 0. Theorems 3.2 and 3.3 and Re-mark 3.3 in [Sa] then yield

λI ´A˚ is Fredholm ðñ λ R σesspA˚q ðñ σpM˘pλqq X iR “ H,indpλI ´A˚q “ dimXu

´pλq ´ dimXu`pλq,

λ R σ0esspA˚q ðñ σpM˘pλqq X iR “ H and dimXu

´pλq “ dimXu`pλq.

Note that here non-zero indices naturally occur. The proofs of these re-sults use Theorems 2.21 and 2.25 and properties of the ordinary differentialequation governed by the matrices

Mλpsq “

ˆ

0 11dpλ´ f

1pu˚psqqq ´ cd

˙

, s P R.

One thus has to study the eigenvalues of M˘pλq (which is easy) to deter-mine the location of the essential spectrum. It then remains the (difficult)task to locate the eigenvalues of A˚. In particular for one space dimension,corresponding tools are discussed in [Sa]. ♦

CHAPTER 3

Sobolev spaces and weak derivatives

Throughout, U Ď Rd is open and non-empty.

3.1. Basic properties

We are looking for properties of C1–functions which can be generalizedto a theory of derivatives suited to Lp spaces. Looking at the theorem ofdominated convergence for instance, one sees that here the basic conceptsshould not be based on pointwise limits. It turns out that integration byparts is an excellent starting point for such a theory.

Let f P C1pUq and ϕ P C8c pUq. Set g0 “ ϕf on suppϕ and extend it by0 to a function g P C1

c pRdq. Then B1g “ pB1ϕqf ` ϕB1f on U since bothsides are trivially equal to 0 on Uz suppϕ. Take a number a ą 0 such thatsupp g Ď p´a, aqd “: Cd and write x “ px1, x

1q. We then deriveż

UpB1fqϕdx “ ´

ż

UfB1ϕdx`

ż

UB1g dx

“ ´

ż

UfB1ϕdx`

ż

Cd´1

ż a

´aB1gpx1, x

1qdx1 dx1

“ ´

ż

UfB1ϕdx`

ż

Cd´1

pgpa, x1q ´ gp´a, x1qqdx1

“ ´

ż

UfB1ϕdx.

Inductively one shows thatż

UpBαfqϕdx “ p´1q|α|

ż

UfBαϕdx, (3.1)

for all f P CkpUq, ϕ P C8c pUq and α P Nd0 with |α| ď k, where

Bα :“ Bα11 ¨ ¨ ¨ B

αdd and |α| :“ α1 ` ¨ ¨ ¨ ` αd.

To imitate (3.1) in a definition, we set

LplocpUq “ tf : U Ñ Cˇ

ˇ f measurable, f|K P LppKq for all compact K Ď Uu

for p P r1,8s. We extend f P LplocpUq by 0 to a measurable function f :

Rd Ñ C without further notice. Convergence in LplocpUq means that therestrictions to K converge in LppKq for all compact K Ď U . Observe thatLppUq Ď LplocpUq Ď L1

locpUq for all 1 ď p ď 8.

Definition 3.1. Let f P L1locpUq, α P Nd0, and p P r1,8s. If there is a

function g P L1locpUq such that

ż

Ugϕdx “ p´1q|α|

ż

UfBαϕdx (3.2)

34

3.1. Basic properties 35

for all ϕ P C8c pUq, then g “: Bαf is called weak derivative of f . We set

Dα,ppUq “ tf P LplocpUqˇ

ˇ D Bαf P LplocpUqu.

Moreover, one defines the Sobolev spaces by

W k,ppUq “ tf P LppUqˇ

ˇ f P Dα,ppUq, Bαf P LppUq for all |α| ď ku

for k P N and endows them with

‖f‖k,p “

$

’

’

’

’

’

&

’

’

’

’

’

%

ˆ

ÿ

0ď|α|ďk‖Bαf‖pp

˙1p

, 1 ď p ă 8,

max0ď|α|ďk

‖Bαf‖8, p “ 8,

where B0f :“ f . We write W 0,ppUq “ LppUq, Bej “ Bj, B “ B1 if d “ 1, and

Dk,p “Ş

0ď|α|ďkDα,p.

As usually, LplocpUq, Dα,ppUq andW k,ppUq are spaces of equivalence classes

modulo the subspace N “ tf : U Ñ Cˇ

ˇ f is measurable, f “ 0 a.e.u.

Remark 3.2. Let α, β P Nd0, p P r1,8s, and k P N.a) We will see in Lemma 3.4 that Bαf is uniquely determined for a.e.

x P U . Formula (3.1) thus implies that CkpUq`N Ď Dα,ppUq for all |α| ď kand that weak and classical derivatives coincide for f P CkpUq.

b) Dα,ppUq is a vector space and the map Bα : Dα,ppUq Ñ LplocpUq islinear.

c) Let f P Dα,ppUq X Dα`β,ppUq. Then Bαf belongs to Dβ,ppUq andBβBαf “ Bα`βf . Hence, if also f P Dβ,ppUq, then Bβf P Dα,ppUq andBαBβf “ Bα`βf “ BβBαf .

Proof. Let ϕ P C8c pUq and f P pDα,ppUqXDα`β,ppUqq Ď LplocpUq. Using(3.2) and Schwarz’ theorem from Analysis 2, we then compute

p´1q|β|ż

UpBαfq Bβϕdx “ p´1q|α|`|β|

ż

Uf BαBβϕdx

“ p´1q|α`β|ż

Uf Bα`βϕdx

“

ż

UpBα`βfqϕdx;

i.e., Bαf P Dβ,ppUq and BβBαf “ Bα`βf . l

d) pW k,ppUq, ‖¨‖k,pq is a normed vector space. A sequence pfnqn converges

in W k,ppUq if and only if pBαfnqn converges in LppUq for each α with |α| ď k.Note that fp1,p “ f

pp ` |∇f |p pp for f PW 1,ppUq and p ă 8.

e) The map

J : W k,ppUq Ñ LppUqm; f ÞÑ pBαfq|α|ďk ,

is a linear isometry, where m is the number of α in Nd0 with |α| ď k andLppUqm has the norm pfjqj “ |pfjpqj |p. Since the p-norm and the 1-norm

3.1. Basic properties 36

on Rm are equivalent, there are constants Ck, ck ą 0 such that

ckÿ

0ď|α|ďk‖Bαf‖p ď ‖f‖k,p ď Ck

ÿ

0ď|α|ďk‖Bαf‖p

for all f PW k,ppUq. ♦

We start with simple, but instructive one-dimensional examples.

Example 3.3. a) Let f P CpRq be such that f˘ :“ f|R˘ belong to C1pR˘q.We then have f P D1,1pRq with

Bf “

"

f 1` on r0,8qf 1´ on p´8, 0q

*

“: g.

For fpxq “ |x|, we thus obtain Bf “ 1R` ´ 1p´8,0q, for instance.Proof. For every ϕ P C8c pRq, we compute

ż

Rfϕ1 dt “

ż 0

´8

f´ϕ1 dt`

ż 8

0f`ϕ

1 dt

“ ´

ż 0

´8

f 1´ϕdt` f´ϕ|0´8 ´

ż 8

0f 1`ϕdt` f`ϕ

1|80

“ ´

ż

Rgϕdt,

since f`p0q “ f´p0q by the continuity of f . l

b) The function f “ 1R` does not belong to D1,1pRq.Proof. Assume there would exist g “ Bf P L1

locpRq. Then we wouldobtain for every ϕ P C8c pRq that

ż

Rgϕdt “ ´

ż

R1R`ϕ

1 dt “ ´

ż 8

0ϕ1ptqdt “ ϕp0q.

Taking ϕ with suppϕ Ď p0,8q, we deduce from Lemma 3.4 below that g “ 0on p0,8q. Similarly, it follows that g “ 0 on p´8, 0q. Hence, g “ 0 and soϕp0q “ 0 for all ϕ P C8c pRq, which is false. l

c) Set fpx, yq “ 1R`pxq for px, yq P R2. Then there exists the weakderivatives B2f “ 0, and thus B1B2f “ B2B2f “ 0. However, the weakderivative B1f does not exist. (See exercises.) ♦

So far we have just used the definition of weak derivatives by duality. Forfurther examples and deeper results one needs mollifiers which we recall anddiscuss next.

Fix a function 0 ď χ P C8pRdq with support Bp0, 1q and χ ą 0 on Bp0, 1q.For x P Rd and ε ą 0, we set

kpxq “1

‖χ‖1χpxq and kεpxq “ ε´dkp1

εxq.

Note that 0 ď kε P C8pRdq, kεpxq ą 0 if and only if |x|2 ă ε, and ‖kε‖1 “ 1.

Let f P L1locpRdq and ε ą 0. We now introduce the mollifier Gε by

Gεfpxq “ pkε ˚ fq pxq “

ż

Bpx,εqkεpx´ yqfpyqdy “

ż

Bp0,εqkεpzqfpx´ zqdz,

(3.3)

3.1. Basic properties 37

for x P Rd or x P U , where we have set fpxq “ 0 for x P RdzU . For a subsetS of Banach space and ε ą 0, we define

Sε “ S `Bp0, εq.

From e.g. Proposition 4.13 in [Sc2] and its proof we recall that

Gεf P C8pRdq, (3.4)

suppGεf Ď Sε for S :“ supp f, Sε is compact if S is compact, (3.5)

‖Gεf‖LppUq ď ‖Gεf‖LppRdq ď ‖f‖p if f P LppUq and 1 ď p ď 8, (3.6)

Gεf Ñ f in LppUq as εÑ 0 if f P LppUq and 1 ď p ă 8 (3.7)

or if p “ 8 and f is uniformly continuous.

Lemma 3.4. Let K Ď U be compact. Then there is a function ψ P C8c pUqsuch that 0 ď ψ ď 1 on U and ψ “ 1 on K. Let g P L1

locpUq satisfyż

Ugϕdx “ 0

for all ϕ P C8c pUq. Then g “ 0 a.e.. In particular, weak derivatives areuniquely defined.

Proof. 1) Let 0 ă δ ă 12 distpBK, BUq. Then K2δ “ K ` Bp0, 2δq is

compact and K2δ Ď U . The function ψ :“ Gδ1Kδ thus belongs to C8c pUq by(3.4) and (3.5). Moreover, (3.3) and (3.6) imply that 0 ď ψpxq ď ‖ψ‖8 ď‖1Kδ‖8 “ 1 for all x P U and

ψpxq “

ż

Bpx,δqkδpx´ yq1Kδpyq dy “ ‖kδ‖1 “ 1

for all x P K. The first claim is shown.2) Assume that g ‰ 0 on a Borel set B Ď U with λpBq ą 0. Theorem 2.20

of [Ru1] yields a compact set K Ď B Ď U with λpKq ą 0. Since ψg P L1pUq,the functions Gεpψgq converge to ψg in L1pUq as εÑ 0 due to (3.7). Hence,there is a nullset N and a subsequence εj Ñ 0 with εj ď δ such thatpGεj pψgqqpxq Ñ gpxq ‰ 0 as j Ñ 8 for each x P KzN . For every x P KzNand j P N, we also deduce

pGεj pψgqqpxq “

ż

Ukεj px´ yqψpyq gpyqdy “ 0

from the assumption, since the function y ÞÑ kεj px ´ yqψpyq belongs toC8c pUq. This is a contradiction.

Recall that Holder’s inequality implies that the map

LppBq ˆ Lp1

pBq Ñ C; pf, gq ÞÑ

ż

Bfg dx, (3.8)

is continuous for all 1 ď p ď 8 and Borel sets B Ď Rd. The next lemma isthe key to many properties of weak derivatives.

Lemma 3.5. Let α P Nd0, p P r1,8s, and ε ą 0.(a) Let f P Dα,ppUq and p ă 8. Then the functions Gεf P C8pUq

converge to f and BαpGεfq tend to Bαf in LplocpUq as εÑ 0. Moreover,

BαpGεfqpxq “ GεpBαfqpxq if x P U, ε ă dpx, BUq.

3.1. Basic properties 38

For all εj Ñ 0 we obtain a subsequence εn :“ εjn Ñ 0 such that Gεnf Ñ f

and BαpGεnfq Ñ Bαf a.e. on U as nÑ8. If f also belongs to Dβ,ppUq forsome β P Nd0, we can take the same εn for all such β.

(b) Let f, g P LplocpUq and fn P Dα,ppUq such that fn Ñ f and Bαfn Ñ g

in LplocpUq as n Ñ 8. Then f is contained in Dα,ppUq and Bαf “ g. Ifthese limits exist in LppUq and for all α with |α| ď k, then f is an elementof W k,ppUq.

Proof. (a) 1) Let ε ą 0 and x P U . If ε ă dpx, BUq, then the functiony ÞÑ ϕε,xpyq :“ kεpx´yq belongs to C8c pUq since suppϕε,x “ Bpx, εq. Usinga corollary to Lebesgue’s theorem and (3.2), we can thus deduce

BαGεfpxq “

ż

UBαxkεpx´ yqfpyq dy “ p´1q|α|

ż

UpBαϕε,xqpyqfpyq dy

“

ż

Uϕε,xpyqpB

αfqpyq dy “ pGεBαfqpxq.

2) Choose a compact subset K Ď U and fix δ ą 0 with Kδ Ď U . Takeε P p0, δs. Note that the integrand of pGεgqpxq is then supported in Kδ forevery x P K and g P L1

locpUq, see (3.3). Hence, (3.7) and part 1) imply that

1KBαpGεfq “ 1KGεpB

αfq “ 1KGεp1KδBαfq ÝÑ 1K1KδB

αf “ 1KBαf

converge in LppKq as εÑ 0. So the asserted convergence in LplocpUq is true.3) For m P N, we define

Km “ tx P Uˇ

ˇ dpx, BUq ě 1m and |x|2 ď mu

These sets are compact andŤ

mPNKm “ U . Let εj Ñ 0. Then, for eachm P N there is a null set Nm Ď Km and a subsequence νm Ď νm´1 suchthat BαGενmpkqfpxq tends to Bαfpxq and Gενmpkqfpxq tends to fpxq for all

x P KmzNm as k Ñ8. By means of a diagonal sequence, one obtains a sub-sequence εn “ εjn Ñ 0 such that BαGεnfpxq Ñ Bαfpxq and Gεnfpxq Ñ fpxqfor x P Uzp

Ť

mPNNmq as n Ñ 8, whereŤ

mPNNm is a null set. Employing

another diagonal sequence, we can achieve this for countably many Bβf atthe same time.

(b) Let ϕ P C8c pUq and S “ suppϕ. Since S is compact, we have fn Ñ fand Bαfn Ñ g in LppSq as nÑ8. From (3.8) on S we deduce that

ż

UfBαϕdx “ lim

nÑ8

ż

UfnB

αϕdx “ p´1q|α| limnÑ8

ż

UpBαfnqϕdx

“ p´1q|α|ż

Ugϕdx.

Hence, f P Dα,ppUq and Bαf “ g. The last assertion then easily follows.

In the next examples we also argue by approximation, but using a differ-ent, more explicit regularisation method.

Example 3.6. Let U “ Bp0, 1q.a) Let d ě 2, 1 ď p ă d, and fpxq “ ln|x|2 for x P Uzt0u. Then f belongs

to W 1,ppUq with

Bjfpxq “xj|x|22

“: gjpxq,

for x ‰ 0 and j P t1, ¨ ¨ ¨ , du. Moreover, f P LqpUqzL8pUq for all q P r1,8q.

3.1. Basic properties 39

Proof. Using polar coordinates and |xj | ď r, we obtain

‖f‖qq “ c

ż 1

0|ln r|q rd´1 dr ă 8,

‖gj‖pp ď c

ż 1

0

rp

r2prd´1 dr “ c

ż 1

0rd´p´1 dr ă 8,

since p ă d. Hence, f P LqpUq and gj P LppUq. Define un P C

8pUq Ď

W 1,ppUq by unpxq “ lnpn´2 ` |x|22q12 for n P N. Observe that Bjunpxq “

pn´2 ` |x|22q´1xj , unpxq Ñ fpxq and Bjunpxq Ñ gjpxq as n Ñ 8 for all

x P Uzt0u. We have the pointwise bounds

|unpxq| ď

#

|fpxq|, n´2 ` |x|22 ď 1,

ln?

2, n´2 ` |x|22 ą 1,, |Bjunpxq| ď gjpxq, x P U.

Lebesgue’s theorem thus yields that un Ñ f and Bjun Ñ fj in LppUq asnÑ8. The assertion then follows from Lemma 3.5(b). l

(b) Let p P r1,8q and β P p1´ dp , 1s. Set upxq “ |x|β2 and fjpxq “ βxj |x|

β´22

for x P Uzt0u and j P t1, . . . , du. Then u P W 1,ppUq and Bju “ fj . (SeeExample 4.18 in [Sc2].) ♦

Proposition 3.7. For 1 ď p ď 8 and k P N. Then W k,ppUq is a Banachspace which is isometrically isomorphic to a closed subspace of a LppUqm forsome m P N. It is separable if p ă 8 and reflexive if 1 ă p ă 8. Moreover,W k,2pUq is a Hilbert space endowed with the scalar product

pf |gqk,2 “ÿ

|α|ďk

ż

UBαf Bαg dx.

Proof. Let pfnqn be a Cauchy sequence in W k,ppUq. Then pBαfnqnis a Cauchy sequence in LppUq for every α P Nd0 with |α| ď k and thusBαfn Ñ gα in LppUq for some gα P L

ppUq as n Ñ 8, where we set f :“ g0.Let ϕ P C8c pUq and |α| ď k. Since fn PW

k,ppUq, we deduceż

UfBαϕdx “ lim

nÑ8

ż

UfnB

αϕdx “ limnÑ8

p´1q|α|ż

UpBαfnqϕdx

“ p´1q|α|ż

Ugαϕdx.

This means that gα is the weak derivative Bαf so that f P W k,ppUq andfn Ñ f in W k,ppUq. Hence, W k,ppUq is a Banach space. We then deducefrom Remark 3.2e) that W k,ppUq is isometrically isomorphic to a subspace ofLppUqm which is closed by e.g. Remark 2.11 in [Sc2]. The remaining claimsnow follow by isomorphy from known results of functional analysis.

We next establish product and chain rules. One can derive many variantsby modifications of the proofs.

Proposition 3.8. (a) Let f, g P D1,1pUq X L8pUq. Then, fg belongs toD1,1pUq X L8pUq and has the derivatives

Bjpfgq “ pBjfqg ` fpBjgq, j P t1, . . . , du. (3.9)

3.1. Basic properties 40

(b) Let 1 ď p ď 8, f PW 1,ppUq and g PW 1,p1pUq. Then, fg is containedin W 1,1pUq and satisfies (3.9).

Proof. 1) Let f, g P D1,1pUq. Set fn “ Gεnf P C8pUq and gn “Gεng P C8pUq with εn Ñ 0 as in Lemma 3.5(a). Fix m P N and takeϕ P C8c pUq and j P t1, . . . , du. Choose an open and bounded set V suchthat suppϕ Ď V Ď V Ď U . Since fn Ñ f and Bjfn Ñ Bjf on L1pV q byLemma 3.5(a), the formulas (3.8) and (3.1) yield

ż

UfgmBjϕdx “ lim

nÑ8

ż

VfngmBjϕdx

“ ´ limnÑ8

ż

VppBjfnqgm ` fnpBjgmqqϕdx

“ ´

ż

UppBjfqgm ` fpBjgmqqϕdx,

so that fgm P D1,1pUq and Bjpfgmq “ pBjfqgm ` fpBjgmq.

2) Let f, g P D1,1pUq X L8pUq and gm as in 1). Note that gm Ñ g andBjgm Ñ Bjg in L1

locpUq as mÑ8. Since f is bounded, we obtainż

UfgBjϕdx “ lim

mÑ8

ż

UfgmBjϕdx

“ limmÑ8

´

„ż

UpBjfqgmϕdx`

ż

UfpBjgmqϕdx

using step 1). In the last line, the second integral converges toş

U fpBjgqϕdx,again because of f P L8pUq. For the first integral we use that gm Ñ g a.e.by Lemma 3.5(a) and that ‖gm‖8 ď ‖g‖8 by (3.6). Lebesgue’s theorem(with the majorant |Bjf | ‖g‖8‖ϕ‖81suppϕ) then implies

ż

UfgBjϕdx “ ´

ż

UppBjfqg ` fpBjgqqϕdx.

Note that pBjfqg` fpBjgq P L1locpUq by our assumptions. Part (a) is shown.

3) Let f P W 1,ppUq and g P W 1,p1pUq. If p P p1,8s we show (3.9) as in

step 2), using (3.8) and that gm, Bjgm converge in Lp1

locpUq by Lemma 3.5(a).If p “ 1, we replace the roles of f and g. Holder’s inequality and (3.9) finallyyield that Bjpfgq is contained in L1pUq.

Proposition 3.9. Let 1 ď p ď 8, j P t1, . . . , du, and f P D1,ppUq.(a) Let f be real-valued and h P C1pRq with h1 P CbpRq. Then h˝f belongs

to D1,ppUq with derivatives

Bjph ˝ fq “ ph1 ˝ fqBjf.

(b) Let V Ď Rd be open and Φ “ pΦ1, ¨ ¨ ¨ ,Φdq : V Ñ U be a diffeomor-phism such that Φ1 and pΦ´1q1 are bounded. Then f ˝ Φ belongs to D1,ppV qwith derivatives

Bjpf ˝ Φq “dÿ

k“1

ppBkfq ˝ Φq BjΦk.

In both results we can replace D1,ppUq by W 1,ppUq, where in (a) we thenalso assume that hp0q “ 0 if λpUq “ 8 and p ă 8.

3.1. Basic properties 41

Proof.1 By Lemma 3.5(a), there are fn P C8pUq such that fn Ñ f and

Bjfn Ñ Bjf in LplocpUq and a.e. as nÑ8.(a) The function h ˝ f belongs to LplocpUq since

|hpfpxqq| ď |hpfpxqq ´ hp0q|` |hp0q| ď ‖h1‖8|fpxq|` |hp0q|for all x P U . It is contained in LppUq if f P LppUq and if hp0q “ 0 in thecase that λpUq “ 8 and p ‰ 8. Let K Ď U be compact. We obtain thatż

K|hpfnpxqq ´ hpfpxqq|dx ď ‖h1‖8

ż

K|fnpxq ´ fpxq| dx ÝÑ 0,

ż

K|h1pfnpxqqBjfnpxq ´ h1pfpxqqBjfpxq| dx

ď ‖h1‖8ż

K|Bjfnpxq ´ Bjfpxq| dx`

ż

K|h1pfnpxqq ´ h1pfpxqq||Bjfpxq|dxÑ 0

as n Ñ 8 where we also used Lebesgue’s theorem and the majorant2‖h1‖8|Bjf | in the last integral. Since h˝fn P C

1pUq, Bjph˝fnq “ ph1˝fnqBjfn

and ph1 ˝ fqBjf belongs to LplocpUq, Lemma 3.5(b) yields assertion (a). Iff PW 1,ppUq, then ph1 ˝ fqBjf P L

ppUq and so h ˝ f PW 1,ppUq.(b) Let B “ Φ´1pAq for an open set A Ď U (or the closure of an open set

whose boundary has measure 0) and g P LppAq. For p ă 8, the transforma-tion rule yields

ż

B|gpΦpxqq|p dx “

ż

A|gpyq|p |detrpΦ´1q1pxqs|dx ď c fpLppAq.

An analogous estimate is also true for p “ 8. Hence, f ˝ Φ belongs toLplocpV q and fn ˝ Φ tends to f ˝ Φ in LplocpV q. We further have

Bjpfn ˝ Φq “dÿ

k“1

ppBkfnq ˝ Φq BjΦk.

The above estimate also implies that pBkfnq˝Φ tends to pBkfq˝Φ in LplocpV qas nÑ8. Since BjΦk is uniformly bounded, Lemma 3.5(b) yields that f ˝Φin contained in D1,ppUq and that it has the asserted derivative. If f P LppUqwe can replace throughout LplocpV q by LppV q.

Corollary 3.10. Let f P D1,1pUq be real-valued. Then the functions f`,f´ and |f | belong to D1,1pUq with

Bjf˘ “ ˘1tfż0uBjf and Bj |f | “`

1tfą0u ´ 1tfă0u

˘

Bjf

for all j P t1, . . . , du. Here one can replace D1,1 by W 1,p for all 1 ď p ď 8.

Proof.2 We employ the map hε P C1pRq given by hεptq :“

?t2 ` ε2 ´

ε ď t for t ě 0 and hεptq :“ 0 for t ă 0, where ε ą 0. Observe that ‖h1ε‖8 “ 1for ε ą 0 and that hεptq Ñ 1r0,8qptqt for t P R as ε Ñ 0. Proposition 3.9

shows that hε ˝ f P D1,1pUq and

ż

UhεpfqBjϕdx “ ´

ż

Uh1εpfqpBjfqϕdx “ ´

ż

tfą0u

fa

f2 ` ε2pBjfqϕdx

1Not shown in the lectures.2Not shown in the lectures.

3.1. Basic properties 42

for each ϕ P C8c pUq. Using the majorants ‖Bjϕ‖81B|f | and ‖ϕ‖81B|Bjf |with B “ suppϕ, we deduce from Lebesgue’s convergence theorem that

ż

Uf`Bjϕdx “ ´

ż

tfą0u

f

|f |pBjfqϕdx “ ´

ż

U1tfą0upBjfqϕdx.

There thus exists Bjf` “ 1tfą0uBjf P L1locpUq. Clearly, Bjf` P L

ppUq if f P

W 1,ppUq. The other claims follow from f´ “ p´fq` and |f | “ f` ` f´.

We briefly discuss two special cases, namely d “ 1 and p “ 8.

Theorem 3.11. Let J Ď R be an open interval, 1 ď p ă 8, and f PLplocpJq. Then f belongs to D1,ppJq if and only if there is a function g P

LplocpJq and a representative of f which is continuous on J and satisfies

fptq “ fpsq `

ż t

sgpτq dτ (3.10)

for all s, t P J . In this case, g “ Bf a.e..

Proof. 1) Let f P D1,ppJq. Take the functions fn “ Gεnf P C8pJq

from Lemma 3.5(a). Then for a.e. t P J and for a.e. t0 P J we have

fptq ´ fpt0q “ limnÑ8

pfnptq ´ fnpt0qq “ limnÑ8

ż t

t0

f 1npτqdτ “

ż t

t0

Bfpτq dτ.

Fixing one t0 and noting that t ÞÑştt0Bfpτqdτ is continuous, we obtain a

continuous representative of f on J which fulfills (3.10) for t P J , s “ t0 andg “ Bf . Subtracting two such equations for any given t, s P J and the fixedt0, we deduce (3.10) with g “ Bf for all t, s P J .

2) If (3.10) is satisfied by some g P LplocpJq, take gn P C8pJq such that

gn Ñ g in LplocpJq as n Ñ 8. For any s P J and n P N, the function

fnptq :“ fpsq `şts gnpτq dτ , t P J , belongs to C8pJq with f 1n “ gn. For

ra, bs Ď J with s P ra, bs, we estimate

‖fn ´ f‖pLppra,bsq “ż b

a

ˇ

ˇ

ˇ

ˇ

ż t

spgnpτq ´ gpτqqdτ

ˇ

ˇ

ˇ

ˇ

p

dt

ď

ż b

a|t´ s|

pp1

ˆż b

a|gnpτq ´ gpτq|

p dτ

˙pp

dt

ď pb´ aq1`pp1 ‖gn ´ g‖pLppra,bsq,

using (3.10) and Holder’s inequality. Hence, fn tends to f in LplocpJq asnÑ8. Lemma 3.5(b) then yields f P D1,ppJq and Bf “ g.

Remark 3.12. a) Let J “ pa, bq for some a ă b in R and f : J Ñ C. Wethen have f P W 1,1pJq if and only if f is absolutely continuous; i.e., for allε ą 0 there is a δ ą 0 such that for all subdivisions a ă α1 ă β1 ă ¨ ¨ ¨ ă

αn ă βn ă b with n P N andřnj“1pβj ´ αjq ď δ we obtain

nÿ

j“1

|fpβjq ´ fpαjq| ď ε.

3.1. Basic properties 43

In this case, f is differentiable for a.e. t P J and the pointwise derivative f 1

is equal a.e. to the weak derivative Bf P L1pJq.3

Proof. The implication “ñ” is true since (3.10) yields that

nÿ

j“1

|fpβjq ´ fpαjq| “nÿ

j“1

ˇ

ˇ

ˇ

ˇ

ˇ

ż βj

αj

Bfpτqdτ

ˇ

ˇ

ˇ

ˇ

ˇ

ď

ż

Ťnj“1pαj ,βjq

|Bfpτq|dτ “: S,

where S Ñ 0 as λpŤnj“1pαj , βjqq Ñ 0.

The other implication “ ð2 and the last assertion is shown in Theo-rem 7.20 of [Ru1] (combined with our Theorem 3.11). l

b) There is a continuous increasing function f : r0, 1s Ñ R with fp0q “ 0and fp1q “ 1 such that f 1ptq “ 0 exists for a.e. t P r0, 1s. Consequently,

1 “ fp1q ‰ fp0q `

ż 1

0f 1pτq dτ “ 0,

and so f is not absolutely continuous, does not belong to W 1,1p0, 1q andviolates (3.10). (See §7.16 in [Ru1].)

c) Let f P D1,1pJq and Bf be continuous. Then f is continuously differ-entiable, since then

fptq ´ fpsq

t´ s“

1

t´ s

ż t

sBfpτqdτ ÝÑ Bfpsq

as tÑ s, thanks to Theorem 3.11. ♦.

Proposition 3.13. Let U be convex. Then W 1,8pUq is isomorphic to

C1´b pUq “ tf P CbpUq

ˇ

ˇ f is Lipschitzu,

and the norm of W 1,8pUq is equivalent to

‖f‖C1´b“ ‖f‖8 ` rf sLip,

where rf sLip is the Lipschitz constant of f .

Proof.4 1) Let f P W 1,8pUq. Take εn Ñ 0 from Lemma 3.5(a). LetK Ď U be compact. For sufficiently large n P N, Lemma 3.5 and (3.6) yield

|BjGεnfpzq| “ |GεnBjfpzq| ď ‖Bjf‖8 ď ‖f‖1,8,

for all j P t1, . . . , du and z P K. Using that Gεnfpxq Ñ fpxq as n Ñ 8 forall x P UzN and a null set N , for all x, y P UzN we thus estimate

|fpxq ´ fpyq| “ limnÑ8

|Gεnfpxq ´Gεnfpyq|

“ limnÑ8

ˇ

ˇ

ˇ

ˇ

ż 1

0∇Gεnfpy ` τpx´ yqq ¨ px´ yqdτ

ˇ

ˇ

ˇ

ˇ

ď ‖f‖1,8 |x´ y|2 .

Hence, f has a representative with Lipschitz constant ‖f‖1,8.

3Note that a Lipschitz continuous function is absolutely continuous and that an ab-solutely continuous function is uniformly continuous.

4Not shown in the lectures.

3.2. Density and embedding theorems 44

2) Let f P C1´b pUq. Take ϕ P C8c pUq, j P t1, . . . , du, and δ ą 0 such that

psuppϕqδ Ď U . For ε P p0, δs the difference quotient 1ε pϕpx ` εejq ´ ϕpxqq

converges uniformly on suppϕ as εÑ 0, and henceˇ

ˇ

ˇ

ˇ

ż

UfBjϕdx

ˇ

ˇ

ˇ

ˇ

“ limεÑ0

ˇ

ˇ

ˇ

ˇ

ż

suppϕfpxq

1

εpϕpx` εejq ´ ϕpxqqdx

ˇ

ˇ

ˇ

ˇ

ď limεÑ0

ż

suppϕ

1

ε|fpy ´ εejq ´ fpyq| |ϕpyq|dy

ď rf sLip‖ϕ‖1.

Since C8c pUq is dense in L1pUq, the map ϕ ÞÑ ´ş

U fBjϕdx has a continuous

linear extension Fj : L1pUq Ñ C. There thus exists a function gj P L8pUq “

L1pUq˚ with ‖gj‖8 “ ‖Fj‖ ď rf sLip such that

´

ż

UfBjϕdx “ Fjpϕq “

ż

Ugjϕdx

for all ϕ P C8c pUq. This means that f has the weak derivative Bjf “ gj PL8pUq. As a result, f PW 1,8pUq and ‖f‖1,8 ď ‖f‖8 ` rf sLip.

In the above proof convexity is only used in part 1). One can extendthe result to other classes of domains, see Proposition 3.28. In the spirit ofRemark 3.12, we mention Rademacher’s theorem (Theorem 5.8.5 in [Ev])which says that a Lipschitz continuous function f is differentiable for a.e.x P U and that the weak derivative Bjf coincides with the pointwise one.

3.2. Density and embedding theorems

In this and the next section we discuss some of the most important theo-rems on Sobolev spaces.

Definition 3.14. For k P N and 1 ď p ă 8, the closure of C8c pUq in

W k,ppUq is denoted by W k,p0 pUq.

Theorem 3.15. Let k P N and p P r1,8q. We then have

W k,p0 pRdq “W k,ppRdq.

Moreover, the set C8pUq XW k,ppUq is dense in W k,ppUq.

Proof.5 We prove the theorem only for k “ 1, the general case can betreated similarly.

1) Let f PW 1,ppRdq. Take any φ P C8pRq such that 0 ď φ ď 1, φ “ 1 onr0, 1s and φ “ 0 on r2,8q. Set

ϕnpxq “ φ`

1n |x|2

˘

(“cut-off function”)

for n P N and x P Rd. We then have ϕn P C8c pRdq, 0 ď ϕn ď 1 and‖Bjϕn‖8 ď ‖φ1‖8 1

n for all n P N, as well as ϕnpxq Ñ 1 for all x P Rdas n Ñ 8. Thus ‖ϕnf ´ f‖p Ñ 0 as n Ñ 8 by Lebesgue’s convergencetheorem. Further, Proposition 3.8 implies that

‖Bjpϕnf ´ fq‖p “ ‖pϕnBjf ´ Bjfq ` pBjϕnqf‖pď ‖ϕnBjf ´ Bjf‖p ` 1

n‖φ1‖8‖f‖p ,

5Not shown in the lectures. The first part is Theorem 4.21 in [Sc2] for k “ 1.

3.2. Density and embedding theorems 45

and the right hand side tends to 0 as nÑ 8 for each j P t1, . . . , du. Givenε ą 0, we can thus fix an index m P N such that ‖ϕmf ´ f‖1,p ď ε. Due

to (3.4) and (3.5), the functions G 1npϕmfq belong to C8c pRdq for all n P N.

Equation (3.7) and Lemma 3.5 further yield that

G 1npϕmfq Ñ ϕmf and BjG 1

npϕmfq “ G 1

nBjpϕmfq Ñ Bjpϕmfq

in LppRdq as nÑ8, for j P t1, . . . , du. So there is an index n P N with

‖G 1npϕmfq ´ ϕmf‖1,p ď ε,

and thus

‖G 1npϕmfq ´ f‖1,p ď 2ε.

2) For the second assertion, we only have to consider the case BU ‰ H.Let f PW 1,ppUq. Set

Un “

x P Uˇ

ˇ |x|2 ă n and dpx, BUq ą 1n

(

for all n P N. Then Un Ď Un Ď Un`1 Ď U , Un is compact andŤ8n“1 Un “ U .

Observe that U “Ť8n“1 Un`1zUn´1, where U0, U´1 :“ H. There are

functions ϕn in C8c pUq such that suppϕn Ď Un`1zUn´1, ϕn ě 0, andř8n“1 ϕnpxq “ 1 for all x P U (see e.g. Theorem 5.6 in [RR]).Fix ε ą 0. As in step 1), for each n P N there is a number δn ą 0 such

that gn :“ Gδnpϕnfq P C8c pUq, supp gn Ď psuppϕnfqδn Ď Un`1zUn´1 and

‖gn ´ ϕnf‖1,p ď 2´nε. Define gpxq “ř8n“1 gnpxq for all x P U . Observe

that on each ball Bpx, rq Ď U this sum is finite. Hence, g P C8pUq. Sincef “

ř8n“1 ϕnf , we further have

gpxq ´ fpxq “8ÿ

n“1

pgnpxq ´ ϕnpxqfpxqq,

for all x P U and n P N. Due to ‖gn ´ ϕnf‖1,p ď 2´nε, this series convergesabsolutely in W 1,ppUq, and

‖f ´ g‖1,p ď

8ÿ

n“1

‖gn ´ ϕnf‖1,p ď ε.

Remark 3.16. a) If U is bounded, then W k,p0 pUq ‰W k,ppUq, see Lemma

6.67 in [RR].b) For ‘not too bad’ BU one can replace in C8pUq by C8pUq in Theo-

rem 3.15, see Theorem 3.27 below.

We now want to study embeddings of Sobolev spaces. We clearly have

W k,ppUq ãÑW j,ppUq, (3.11)

W k,ppUq ãÑW j,qpUq if λpUq ă 8, (3.12)

for k ě j ě 0, 1 ď q ď p ď 8. (Recall that we have set W 0,ppUq “ LppUqfor 1 ď p ď 8.) The embedding X ãÑ Y means that there is an injectivemap J P BpX,Y q. Writing c “ ‖J‖, one obtains ‖f‖Y ď c ‖f‖X if oneidentifies Jf with f .

3.2. Density and embedding theorems 46

Theorem 3.17 (Sobolev, Morrey). Let k P N and 1 ď p ă 8. We havethe following embeddings.

(a) If kp ă d, then

p˚ :“pd

d´ kpP pp,8q and W k,ppRdq ãÑ LqpRdq for all q P rp, p˚s.

(b) If kp “ d, then

W k,ppRdq ãÑ LqpRdq for all q P rp,8q.

(c) If kp ą d, then there are either j P N0 and β P p0, 1q such thatk ´ d

p “ j ` β or k ´ dp P N. In the latter case we set j :“ k ´ d

p ´ 1 P N0

and take any β P p0, 1q. Then

W k,ppRdq ãÑ Cj`β0 pRdq

for all x, y P Rd, |α| ď j, a constant c ą 0, and

Cj`β0 pRdq :“ tu P CjpRdqˇ

ˇ Bαupxq Ñ 0 as |x|2 Ñ8 and Bαu is β–Holder

continuous on Rd, for all 0 ď |α| ď ju.

In parts (a) and (b), the embedding J is just the inclusion map, and inpart (c) the function Jf is the continuous representative of f .

We rephrase the above results in a slightly modified way using the ‘effec-tive regularity index’ k ´ d

p of W k,p.

Corollary 3.18. Let k P N, j P N0, and p P r1,8q. We have thefollowing embeddings.

(a) If q P rp,8q and k ´ dp ě j ´ d

q , then

W k,ppRdq ãÑW j,qpRdq.

(b) If q P rp,8q and k ´ dp “ j, then

W k,ppRdq ãÑW j,qpRdq.

(c) If β P p0, 1q and k ´ dp “ j ` β, then

W k,ppRdq ãÑ Cj`β0 pRdq.

Proof of Corollary 3.18. (a) If pk´jqp “ d, then Theorem 3.17(b)yields that

W k´j,ppRdq ãÑ LqpRdq (3.13)

for all q P rp,8q. If pk ´ jqp ą d, then W k´j,ppRdq ãÑ LppRdq by (3.11)and W k´j,ppRdq ãÑ L8pRdq by Theorem 3.17(c). Hence the interpolationinequality (3.14) below implies the embedding (3.13) for all q P rp,8s in thiscase. Let pk´ jqp ă d. By assumption, we have p ď q ď pdpd´pk´ jqpq´1,so that (3.13) for these q is a consequence of Theorem 3.17(a). Applying theembedding (3.13) to Bαf PW k´j,ppRdq for all |α| ď j and f PW k,ppRdq, wededuce

‖Bαf‖q ď c ‖Bαf‖k´j,p ď c ‖f‖k,p.So (a) is true. Part (b) follows from (a), and (c) from Theorem 3.17(c).

3.2. Density and embedding theorems 47

Example 3.19. Let d ě 2, α P p0, 1´ 1dq and ϕ P C8c pRdq with suppϕ Ď

Bp0, 34q and ϕ “ 1 on Bp0, 12q. We define

fpxq :“

#

ϕpxqp´ ln|x|2qα, 0 ă |x|2 ď 34,

0, |x|2 ą 34 or x “ 0.

Then f PW 1,dpRdqzL8pRdq; i.e., Theorem 3.17(b) is sharp if d ě 2.Proof. Arguing as in Example 3.3c), one sees that f P LppRdq for all

p ă 8, f R L8pRdq and that, for all j P t1, . . . , du, we have

Bjfpxq “ pBjϕpxqqp´ ln|x|2qα ´ αϕpxqp´ ln|x|2qα´1 xj|x|22

for 0 ă |x|2 ă 34 and Bjfpxq “ 0 otherwise. Using polar coordinates, wefurther estimate

ˆż

Rd|Bjf |d dx

˙1d

ď c ‖Bjϕ‖8

˜

ż 34

0pln rqαdrd´1 dr

¸1d

` c ‖ϕ‖8

˜

ż 34

0

pln rqpα´1qd

rdrd´1 dr

¸1d

ď c` c

˜

ż 34

0

dr

rpln rqp1´αqd

¸1d

ă 8

for some constants c ą 0, since p1´ αqd ą 1. l

For the proof of Theorem 3.17 we set xj “ px1, . . . , xj´1, xj`1, . . . , xdq P

Rd´1 for all x P Rd, j P t1, . . . , du and d ě 2. We start with a lemma.

Lemma 3.20. Let d ě 2 and f1, . . . , fd P Ld´1pRd´1q X CpRd´1q. Setfpxq “ f1px

1q ¨ . . . ¨ fdpxdq for x P Rd. We then have f P L1pRdq and

‖f‖L1pRdq ď ‖f1‖Ld´1pRd´1q ¨ . . . ¨ ‖fd‖Ld´1pRd´1q.

Proof.6 If d “ 2, then Fubini’s theorem shows thatż

R2

|fpxq| dx “ż

R

ż

R|f1px2q||f2px1q|dx1 dx2 “ ‖f1‖1‖f2‖1,

as asserted. Assume that the assertion holds for some d P N with d ě 2.Take f1, . . . , fd`1 P L

dpRdq X CpRdq. Write y “ px1, . . . , xdq P Rd andx “ py, xd`1q P Rd`1. For a.e. xd`1 P R, the maps yj ÞÑ |fjpyj , xd`1q|d are

integrable on Rd´1 for each j P t1, . . . , du due to Fubini’s theorem. Fix sucha xd`1 P R and write

fpy, xd`1q :“dź

j“1

fjpxjq.

Using Holder’s inequality, we obtainż

Rd|fpy, xd`1q|dy “

ż

Rd|fpy, xd`1q| |fd`1pyq|dy

ď ‖fd`1‖LdpRdqˆż

Rd|fpy, xd`1q|d

1

dy

˙1d1

.

6Not shown in the lectures.

3.2. Density and embedding theorems 48

We set gjpyjq “ |fjpyj , xd`1q|d

1

for j P t1, . . . , du and x P Rd`1. Since

d1pd´1q “ d, the maps gj belong to Ld´1pRd´1q and the induction hypothesisyieldsż

Rd|fpy, xd`1q|d

1

dy “

ż

Rdg1py

1q ¨ . . . ¨ gdpydq dy ď ‖g1‖d´1 ¨ . . . ¨ ‖gd‖d´1

“

dź

j“1

ˆż

Rd´1

|fjpyj , xd`1q|d dy

˙1d´1

.

Integrating over xd`1 P R, we thus arrive atż

Rd`1

|f |dx ď ‖fd`1‖dż

R

dź

j“1

ˆż

Rd´1

|fjpxjq|d dy

˙1d´1

d´1d

dxd`1.

Applying the d-fold Holder inequality to the xd`1-integral, we conclude that

ż

Rd`1

|f |dx ď ‖fd`1‖ddź

j“1

˜

ż

R

ˆż

Rd´1

|fjpxjq|d dy

˙1d¨d

dxd`1

¸

1d

“ ‖f1‖d ¨ . . . ¨ ‖fd`1‖d.

Recall from Analysis 3 that for f P LppUq X LqpUq and r P rp, qs with1 ď p ă q ď 8, we have

‖f‖r ď ‖f‖θp ‖f‖1´θq ď θ‖f‖p ` p1´ θq‖f‖q, (3.14)

where θ P r0, 1s is given by 1r “

θp `

1´θq and we also used Young’s inequality

from Analysis 1+2.

Proof of Theorem 3.17.7 We only prove the case k “ 1, the rest canbe done by induction, see e.g. §5.6.3 in [Ev]. Since W 1,ppRdq ãÑ LppRdq, inview of (3.14) for assertion (a) it suffices to show

W 1,ppRdq ãÑ Lp˚

pRdq.1) Let f P C1

c pRdq. Let first p “ 1 ă d, whence p˚ “ dd´1 . For x P Rd and

j P t1, . . . , du, we then obtain

|fpxq| “ˇ

ˇ

ˇ

ˇ

ż xj

´8

Bjfpx1, . . . , xj´1, t, xj`1, . . . , xdq dt

ˇ

ˇ

ˇ

ˇ

ď

ż

R|Bjfpxq| dxj ,

|fpxq|d ďdź

j“1

ż

Rd|Bjfpxq| dxj .

Setting gjpxjq “ p

ş

R|Bjfpxq|dxjq1d´1 , we deduce

|fpxq|dd´1 ď

dź

j“1

gjpxjq.

After integration over x P Rd, Lemma 3.20 yields

‖f‖dd´1

Ldd´1 pRdq

ď

ż

Rdg1px

1q ¨ . . . ¨ gdpxdq dx ď

dź

j“1

‖gj‖Ld´1pRd´1q

7Not shown in the lectures.

3.2. Density and embedding theorems 49

“

dź

j“1

ˆż

Rd´1

ż

R|Bjfpxq|dxj dxj

˙1d´1

,

‖f‖L

dd´1 pRdq

ď

dź

j“1

‖Bjf‖1d

L1pRdq ď ‖|∇f |1‖1 ď ‖f‖1,1. (3.15)

2) Next, let p P p1, dq and p˚ “ pdd´p . Set t˚ :“ d´1

d p˚ “ d´1d´pp ą 1. An

elementary calculation shows that pt˚ ´ 1qp1 “ t˚dd´1 “ p˚. Set

g “ f |f |t´1 “ fpffqt´12

for t ą 1. We compute

Bjg “ Bjf |f |t´1 ` ft´ 1

2pffq

t´12´1

`

pBjfqf ` fpBjfq˘

“ Bjf |f |t´1 ` pt´ 1qf |f |t´3 RepfBjfq,

|g| “ |f |t, |Bjg| ď t|Bjf ||f |t´1.

Applying (3.15) to g, we estimate

‖f‖t tdd´1

“

ˆż

Rd|f |t

dd´1 dx

˙d´1d

“

ˆż

Rd|g|

dd´1 dx

˙d´1d

ď

dź

j“1

‖Bjg‖1d1 ď

dź

j“1

t1d

ˆż

Rd|Bjf | |f |t´1 dx

˙1d

ď tdź

j“1

´

ż

Rd|Bjf |p dx

¯1dp

ˆż

Rd|f |pt´1qp1 dx

˙1p1d

ď tdź

j“1

‖|∇f |p‖1dp ‖f‖

t´1d

pt´1qp1 “ t ‖|∇f |p‖p ‖f‖t´1pt´1qp1 ,

where we used Holder’s inequality. For t “ t˚, we use the properties of tstated above and obtain

‖f‖p˚ ď pd´ 1

d´ p‖|∇f |p‖p ď p

d´ 1

d´ p‖f‖1,p. (3.16)

This estimate can be extended to all f P W 1,ppRdq by density (see Theo-rem 3.15). Hence, the inclusion map is the required embedding in (a).

3) Let p “ d, f P C1c pRdq, and t ą 1. Then p1 “ d

d´1 , and step 2) yields

‖f‖t dd´1

ď t1t ‖f‖1´ 1

t

pt´1q dd´1

‖|∇f |p‖1td ď c

´

‖f‖pt´1q d

d´1` ‖|∇f |p‖d

¯

(3.17)

using Young’s inequality from Analysis 1+2. For t “ d, this estimate gives

f P Ld2

d´1 pRdq and

‖f‖ d2

d´1

ď c ‖f‖1,d.

Here and below the constants c ą 0 do not depend on f . For q P pd, d dd´1q,

inequality (3.14) further yields

‖f‖q ď c p‖f‖d ` ‖f‖ d2

d´1

q ď c ‖f‖1,d.

3.2. Density and embedding theorems 50

Now, we can apply (3.17) with t “ d` 1 and obtain

‖f‖ d2`dd´1

ď c p‖f‖ d2

d´1

` ‖|∇f |p‖dq ď c ‖f‖1,d.

As above, we see that f P LqpRdq for d ď q ď dd`1d´1 . We can now iterate

this procedure with tn “ d` n and obtain

‖f‖q ď cpqq‖f‖1,p

for all q ă 8. As above, assertion (b) follows by approximation.4) Let p ą d, f P C1

c pRdq, Qprq “ r´ r2 ,

r2 sd for some r ą 0, and x0 P Qprq.

Set Mprq “ r´dş

Qprq f dx. We further put β :“ 1 ´ dp P p0, 1q. Using

|x ´ x0|8 ď r for x P Qprq, the transformation y “ tpx ´ x0q and Holder’sinequality, we compute

|fpx0q ´Mprq| “

ˇ

ˇ

ˇ

ˇ

ˇ

r´dż

Qprqpfpx0q ´ fpxqqdx

ˇ

ˇ

ˇ

ˇ

ˇ

“ r´d

ˇ

ˇ

ˇ

ˇ

ˇ

ż

Qprq

ż 0

1

d

dtfpx0 ` tpx´ x0qqdtdx

ˇ

ˇ

ˇ

ˇ

ˇ

ď r´dż

Qprq

ż 1

0|∇fpx0 ` tpx´ x0qq ¨ px´ x0q| dt dx

ď r1´d

ż 1

0

ż

Qprq|∇fpx0 ` tpx´ x0qq|1 dx dt

“ r1´d

ż 1

0

ż

tpQprq´x0q|∇fpx0 ` yq|1 dy t´d dt

ď r1´d

ż 1

0

„ż

Rd|∇fpx0 ` yq|p1 dy

1p

λptpQprq ´ x0qq1p1 t´d dt

ď cr1´d‖|∇f |p‖pż 1

0rdp1 t

dp1´d

dt

“ Cr1´ d

p ‖|∇f |p‖p

for constants C, c ą 0 only depending on d and p, using also that dp1´d ą ´1

due to p ą d. A translation then gives

ˇ

ˇ

ˇfpx0 ` zq ´ r

´d

ż

z`Qprqfpyqdy

ˇ

ˇ

ˇď Crβ‖|∇f |p‖p

for all z P Rd. Taking x “ z, x0 “ 0, r “ 1, and using Holder’s inequality,we thus obtain

|fpxq| ďˇ

ˇ

ˇfpxq ´

ż

x`Qp1qf dy

ˇ

ˇ

ˇ`

ˇ

ˇ

ˇ

ż

x`Qp1qf dy

ˇ

ˇ

ˇ

ď C‖|∇f |p‖p ` ‖f‖p ď c ‖f‖1,p (3.18)

for all x P Rd, where c only depends on d and p. Given x, y P Rd, we find acube Q of side length |x ´ y|8 “: r such that x, y P Q and Q is parallel to

3.2. Density and embedding theorems 51

the axes. Hence,

|fpxq ´ fpyq| ďˇ

ˇ

ˇfpxq ´ r´d

ż

Qf dy

ˇ

ˇ

ˇ`

ˇ

ˇ

ˇr´d

ż

Qf dy ´ fpyq

ˇ

ˇ

ˇ

ď 2C‖|∇f |p‖p |x´ y|β8 ď 2C‖|∇f |p‖p |x´ y|β2 .

If f P W 1,ppRdq, then there are fn P C1c pRdq converging to f in W 1,ppRdq.

By (3.18), fn is a Cauchy sequence in C0pRdq. Hence, f has a representative

f P C0pRdq such that fn Ñ f uniformly as n Ñ 8. So the above estimatesimply that

‖f‖8 ` supx‰y

|fpxq ´ fpyq||x´ y|β2

ď c ‖f‖1,p.

The map f ÞÑ f is the required embedding.

Remark 3.21. Theorem 3.17 remains valied on any open set U instead of

Rd if we replace W k,p by W k,p0 . In fact, we obtain the desired estimate for

f P C8c pUq if we apply Theorem 3.17 to the 0–extension of f . The result

for f PW k,p0 pUq then follows by density. ♦

Corollary 3.22. Let8 U Ď Rd be open and bounded. We then havePoincare’s inequality

ż

U|∇u|pp dx ě δ

ż

U|u|p dx (3.19)

for some δ ą 0 and all u PW 1,p0 pUq and p P r1,8q.

Proof. For p P r1, dq, the estimate (3.19) follows from (3.16) since

Lp˚

pUq ãÑ LppUq. Let p P rd,8q. The case p “ d “ 1 is easy to check. For

the other cases, fix r P pp,8q and u P W 1,p0 pUq. Then (3.14), (3.15) and

W 1,p0 pUq ãÑ LrpUq imply

up ď cεu1 ` εur ď cε|∇u|11 ` cεu1,pď cε|∇u|pp ` cεup ` cε|∇u|pp

for all ε ą 0 and some constants cε, c ą 0 independent of u, where c doesnot depend on ε ą 0. Choosing a small ε, we derive (3.19).

Remark 3.23. We can show (parts of) Theorem 3.17(a) and (c) for thecase p “ 2 using the Fourier transform. We first recall some basic facts. Forf P L1pRdq, we define Fourier transform by

fpξq “ pFfqpξq :“1

p2πqd2

ż

Rde´i ξ¨xfpxqdx, ξ P Rd,

where the dot denotes the scalar product in Rd. Plancherel’s theorem saysthat F can be extended from L1pRdq X L2pRdq to an isometric bijectionF : L2pRdq Ñ L2pRdq. Moreover, F is a bijection on the Schwartz space Sdwith inverse given by pF´1gqpyq “ pFgqp´yq. See e.g. Proposition 5.10 andTheorem 5.11 in [Sc1]. For Sobolev spaces we have the crucial identity

W k,2pRdq “ tu P L2pRdq | |ξ|k2 pu P L2pRdqu (3.20)

8This result was not part of the lectures.

3.2. Density and embedding theorems 52

for k P N, where |ξ|k2 stands for the function ξ ÞÑ |ξ|k2 (and analogously forsimilar expressions). Moreover, the norm of W k,2pRdq is equivalent to theone given by u22 ` |ξ|

k2 pu

22 “ pu

22 ` |ξ|

k2 pu

22 and

FpBαuq “ iξαpu for u PW k,2pRdq, |α| ď k. (3.21)

See e.g. Theorem 5.21 in [Sc1]. We further use the Hausdorff–Young in-equality

Ffq ď c fq1 for q P r2,8s, f P Lq1

pRdq, (3.22)

see e.g. Satz V.2.10 in [We].We come to the announced proofs. Let f P C8c pRdq and k P N. Then

pf P Sd Ď L1pRdq.1) Let k ą d2. Using the Fourier inversion formula, (3.22) for q “ 8,

Holder’s inequality, (3.20) and polar coordinates, we compute

f8 “ F´1Ff8 ď p2πq´d2 p1` |ξ|k2q

´1p1` |ξ|k2qpf1

ď p2πq´d2 p1` |ξ|k2q

´12 p1` |ξ|k2qpf2

ď c fk,2

ˆż 8

0

rd´1

p1` rkq2dr

˙12

ď c fk,2

for constants c ą 0, since 2k ą d. Next, let g PW k,2pRdq. By Theorem 3.15,there are fn P C

8c pRdq Ď C0pRdq which tend to g in W k,2pRdq and thus in

L2pRdq. The above estimate implies that pfnq is a Cauchy sequence for thesup-norm. The limit of pfnq in C0pRdq is then a representative of its limit g inL2pRdq. In particular, g also satisfies the above estimate and we have shownTheorem 3.17(c) for p “ 2, j “ 0 and β “ 0; i.e., W k,2pRdq ãÑ C0pRdq.

2) Let k ă d2 and 2 ď q ă 2˚ “ 2dpd´ 2kq´1. The latter is equivalent to

1

qą

1

2´k

d.

To apply Holder’s inequality, we define the number s P p2,8s by

1

s“

1

q1´

1

2“

1

2´

1

q. Hence,

1

săk

d.

As in part 1), we estimate

fq ď c p1` |ξ|k2q´1p1` |ξ|k2q

pfq1 ď c p1` |ξ|k2q´1s p1` |ξ|

k2qpf2

ď c fk,2

ˆż 8

0

rd´1

p1` rkqsdr

˙1s

ď c fk,2,

where we have used that sk ą d by the above relations between the expo-nents. One can again conclude that W k,2pRdq ãÑ LqpRdq, which is Theo-rem 3.17(a) for p “ 2 and q ă 2˚.9 ♦

Problem: How to extend the Sobolev embedding to W k,ppUq?This problem can be solved by an extension operator ; i.e., a map E P

BpW k,ppUq,W k,ppRdqq satisfying Eu “ u on U . We state a rather advancedtheorem that provides such an operator for open sets U Ď Rd with a compactLipschitz boundary. This means that each x P BU has a neighborhood Vx in

9One can show the limiting case q “ 2˚ by a refinement of the above argument.

3.2. Density and embedding theorems 53

Rd such that there is a Lipschitz function fx : Rd´1 Ñ R such that (possiblyafter a rotation) we have U X Vx “ ty P Vx

ˇ

ˇ yd ă fxpy1, ¨ ¨ ¨ , yd´1qu. Dueto compactness, we can then cover BU by finitely many such Vx, cf. §4.9 in[AF]. Clearly, compact boundaries of type Ck as defined in Section 2.3 arealso compact Lipschitz boundaries. We further need the upper and lowerhalf spaces

Rd˘ “ tpx1, yqˇ

ˇx1 P Rd, y ż 0u.

Theorem 3.24 (Stein extension theorem). Let U have a compact Lipschitzboundary or let U “ Rd`. Let k P N and p P r1,8s. Then there exists

an operator Ek,p in BpW k,ppUq,W k,ppRdqq with Ek,pu “ u on U . Theseoperators are restrictions of each other. We thus write EU for all of them.

An even more general result is proved in Theorem VI.5 of [St]. A sketchof the proof is given in §5.25 of [AF].

Remark 3.25. We sketch a proof of the extension theorem for W 1,ppUqwith p P r1,8s if BU is compact and of class C1.10 One needs a Ck–boundaryto prove the corresponding result on W k,ppUq by the method below.

1) For any open U , p P r1,8s and k P N0, we define the 0-extensionoperator E0 : LppUq Ñ LppRdq by E0fpxq “ fpxq for x P U and E0fpxq “ 0for x P RdzU and the restriction operator RU : W k,ppRdq Ñ W k,ppUq byRUf “ f|U . Clearly, E0 is a linear isometry and RU is a linear contraction.

Let f PW 1,ppRd´q X C1pRd´q. We write elements in Rd˘ as py, tq. Define

E´fpy, rq “

#

fpy, tq, py, tq P Rd´,4fpy,´ t

2q ´ 3fpy,´tq, py, tq P Rd`.

Note that E´f belongs to C1pRdq. One can check that ‖E´f‖W 1,ppRdq ď

c ‖f‖W 1,ppRd´qfor a constant c ą 0.

2) We show that W 1,ppRd´qXC1pRd´q is dense in W 1,ppRd´q, so that E´ can

be extended to an extension operator onW 1,ppRd´q. In fact, let f PW 1,ppRd´qand ε ą 0. Theorem 3.15 yields a function g P C8pH´q XW 1,ppRd´q with

‖f ´ g‖1,p ď ε. Setting gnpy, tq “ gpy, t´ 1nq for t ď 0, y P Rd´1 and n P N,

we define maps gn in C1pRd´q XW 1,ppRd´q. Observe that

Bαgn “ RRd´SnE0B

αg

for 0 ď |α| ď 1, where Sn P BpLppRdqq is given by Snhpy, tq “ hpy, t ´ 1nq

for h P LppRdq. One can see that SnhÑ h in LppRdq as in Example 4.12 of[Sc2]. Hence, gn converges to g in W 1,ppRd´q implying the claim.

3) Since BU P C1, there are bounded open sets U0, U1, . . . , Um Ď Rd suchthat U Ď U0 Y ¨ ¨ ¨ Y Um, U0 Ď U and BU Ď U1 Y ¨ ¨ ¨ Y Um, as well as adiffeomorphism Ψj : Uj Ñ Vj such that Ψ1j and pΨ´1

j q1 are bounded and

ΨjpUj X Uq Ď Rd´ and ΨjpUj X BUq Ď Rd´1 ˆ t0u, for each j P t1, . . . ,mu.

Theorem 5.6 in [RR] provides us with functions 0 ď ϕj P C8c pRdq with

suppϕj Ď Uj for all j “ 0, 1, . . . ,m andřmj“0 ϕjpxq “ 1 for all x P U .

10Not shown in the lectures.

3.2. Density and embedding theorems 54

Let j P t1, . . . ,mu. Set Sjgpyq “ gpΨ´1j pyqq for y P Rd´XVj and Sjgpyq “ 0

for y P Rd´zVj , where g P W 1,ppUj X Uq. For h P W 1,ppRdq, set Sjhpxq “

hpΨjpxqq for x P Uj and Sjhpxq “ 0 for x P RdzUj . Take any ϕj P C8c pRdq

with supp ϕj Ď Uj and ϕj “ 1 on suppϕj (see Lemma 3.4). Let f PW 1,ppUq.We now define

E1f “ E0ϕ0f `mÿ

j“1

ϕjSjE´Sj`

RpUjXUqpϕjfq˘

.

Using part 2) and Propositions 3.8 and 3.9, we see that E1 belongs toBpW 1,ppUq,W 1,ppRdqq. Let x P U . If x P Uk for some k P t1, . . . ,mu,we have Ψkpxq P Rd´. If x R Uj , then ϕjpxq “ 0. Thus

E1fpxq “ ϕ0pxqfpxq `ÿ

j“1,...,mxPUj

ϕjpxqpϕjfqpΨ´1j pΨjpxqqq

“

mÿ

j“0

ϕjpxqfpxq “ fpxq.

If x P U0zpU1 Y ¨ ¨ ¨ Y Umq, we also have E1fpxq “ ϕ0fpxq “ fpxq. ♦

We can now easily extend and improve the above results.

Theorem 3.26 (Sobolev–Morrey on U). Let U have a compact Lipschitzboundary or let U “ Rd`. Theorem 3.17 and Corollary 3.18 then remain

true if we replace Rd by U and Cj0pRdq by

Cj0pUq “ tf P CjpUq

ˇ

ˇ Bαf has a continuous extension to BU, Bαfpxq Ñ 0

as |x|2 Ñ8 if U is unbounded, for all 0 ď |α| ď ju.

Proof. Consider e.g. Theorem 3.17(a). We have the embedding

J : W k,ppRdq ãÑ Lp˚

pRdqgiven by the inclusion. Thanks to Theorem 3.24, the map

RUJEU : W k,ppUq Ñ Lp˚

pUq

is continuous and injective. The other assertions are proved similarly.

Theorem 3.27. Let U have a compact Lipschitz boundary or let U “ Rd`,k P N and p P r1,8q. Then

W k,ppUq X C8pUq is dense in W k,ppUq,

where C8pUq contains all f P C8pUq such that f and all its derivatives cancontinuously be extended to BU .

Proof. Let f P W k,ppUq. Then EUf belongs to W k,ppRdq by The-orem 3.24. Theorem 3.15 yields functions gn in C8c pRdq that converge toEUf in W k,ppRdq. Hence, RUgn is contained in W k,ppUqXC8pUq and tendsto f “ RUEUf in W k,ppUq as nÑ8.

Proposition 3.28. Let U have a compact C1–boundary. Then W 1,8pUqis isomorphic to

C1´b pUq “ tf P CbpUq

ˇ

ˇ f is Lipschitzu,

3.2. Density and embedding theorems 55

and the norm of W 1,8pUq is equivalent to

‖f‖C1´b“ ‖f‖8 ` rf sLip,

where rf sLip is the Lipschitz constant of f .

Proof. The extension operator E1 from Remark 3.25 also is continu-ous from C1´

b pUq to C1´b pRdq. Owing to Proposition 3.13 we have an iso-

morphism J from W 1,8pRdq to C1´b pRdq given by choosing the continuous

representative. Then RUJE1 is the asserted isomorphism.

We continue with one of the main compactness results in analysis.

Theorem 3.29 (Rellich-Kondrachov). Let U Ď Rd be bounded and openwith a Lipschitz boundary BU , k P N and 1 ď p ă 8. Then the followingassertions hold.

(a) If kp ď d and 1 ď q ă p˚ “ dpd´kp P pp,8s, then the inclusion map

J : W k,ppUq ãÑ LqpUq

is compact. (For instance, let q “ p.)(b) If k ´ d

p ą j P N0, then the embedding

J : W k,ppUq ãÑ CjpUq

is compact, where Jf is the continuous representative.

Recall that a compact embedding J : Y ãÑ X means that any boundedsequence pynq in Y has a subsequence such that pJynj qj converges in X.

Proof of Theorem 3.29.11 We prove the result only for k “ 1 (andthus j “ 0), see e.g. Theorem 6.3 of [AF] for the other cases. Part (b)follows from the Arzela-Ascoli theorem since Theorem 3.26 gives constantsβ, c ą 0 such that |fpxq ´ fpyq| ď c |x ´ y|β and |fpxq| ď c for all x, y P Uand f PW 1,ppUq with ‖f‖1,p ď 1, where p ą d.

In the case p ă d, take fn P W1,ppUq with ‖fn‖1,p ď 1 for all n P N. Fix

an open bounded set V Ď Rd containing U . Lemma 3.4 yields a functionϕ P C8c pV q which is equal to 1 on U . Let EU be given by Theorem 3.24. Setgn “ ϕEUfn P W

1,ppRdq. These functions have support V and ‖gn‖1,p ď

c ϕ1,8‖EU‖ “: M for all n P N. Take q P r1, p˚q and θ P p0, 1s with1q “

θ1 `

1´θp˚ . Inequality (3.14) and Theorem 3.17 yield that

‖fn ´ fm‖LqpUq ď ‖gn ´ gm‖LqpV q ď ‖gn ´ gm‖θL1pV q‖gn ´ gm‖1´θLp˚ pV q

ď c ‖gn ´ gm‖θL1pV qp‖gn‖1´θ1,p ` ‖gm‖1´θ

1,p q

ď 2cM1´θ‖gn ´ gm‖θL1pV q

for all n,m P N. So it suffices to construct a subsequence of gn whichconverges in L1pV q. For x P V , n P N and ε ą 0, we compute

|gnpxq ´Gεgnpxq| “ˇ

ˇ

ˇ

ˇ

ż

Rdkεpx´ yqpgnpxq ´ gnpyqqdy

ˇ

ˇ

ˇ

ˇ

11Not shown in the lectures.

3.2. Density and embedding theorems 56

ď ε´dż

Bpx,εqkp1

ε px´ yqq |gnpxq ´ gnpyq|dy

“

ż

Bp0,1qkpzq|gnpxq ´ gnpx´ εzq|dz

“

ż

Bp0,1qkpzq

ˇ

ˇ

ˇ

ˇ

ż ε

0

d

dtgnpx´ tzq dt

ˇ

ˇ

ˇ

ˇ

dz

ď

ż

Bp0,1qkpzq

ż ε

0|∇gnpx´ tzq ¨ z| dt dz

ď

ż ε

0

ż

Bp0,1qkpzq |∇gnpx´ tzq|2 dz dt

“

ż ε

0

ż

Bpx,tqt´dkp1

t px´ yqq |∇gnpyq|2 dy dt

“

ż ε

0‖kt ˚ |∇gn|2‖1 dt ď ε sup

0ďtďε‖kt‖1 ‖|∇gn|2‖1

ď cε ‖|∇gn|p‖p ď cMε,

where we have used the transformations z “ 1ε px ´ yq and y “ x ´ tz, as

well as Fubini’s theorem, Young’s inequality, (3.6) and LppV q ãÑ L1pV q. Wethus obtain

‖gn ´Gεgn‖L1pV q ď cλpV qMε “: Cε (3.23)

for all n P N and ε ą 0. On the other hand, the definition of Gεgn yields

|Gεgnpxq| ď ‖kε‖8 ‖gn‖L1pV q, and |∇Gεgnpxq| ď ‖∇kε‖8 ‖gn‖L1pV q

for all x P V and n P N and each fixed ε ą 0. The Arzela-Ascoli theoremnow implies that the set Fε :“ tGεgn

ˇ

ˇn P Nu is relatively compact in CpV q

for each ε ą 0, and thus in L1pV q since CpV q ãÑ L1pV q. Let δ ą 0 be givenand fix ε “ δ

2C . Then there are indeces n1, . . . , nl P N such that

Fε Ďďl

j“1BL1pV qpGεgnj ,

δ2q “:

ďl

j“1Bj .

Hence, given n P N, there is an index nj such that Gεgn P Bj . The estimates(3.23) and (3.6) then yield

‖gn´Gεgnj‖L1pV q ď ‖gn´Gεgn‖L1pV q`‖Gεpgn´ gnj q‖L1pV q ď Cε` δ2 “ δ.

We have shown that, for each δ ą 0, the set G :“ tgnˇ

ˇn P Nu is covered by

finitely many open balls Bj of radius δ2; i.e., G is totally bounded in L1pV q.Thus G contains a subsequence converging in L1pV q (see e.g. Corollary 1.39in [Sc2]). In the case p “ d one simply replaces p˚ by any r P pq,8q.

Remark 3.30. a) Theorem 3.29 is wrong for unbounded domains, in gen-eral. In fact, let k P N, p P r1,8q and define the functions fn “ fp¨ ´ nq inW k,ppRq for any function 0 ‰ f P C8pRq with supp f Ď p´12, 12q. Then‖fn‖k,p and ‖fn ´ fm‖q ą 0 do not depend on n ‰ m in N so that pfnq is

bounded in W k,ppRq and has no subsequence which converges in Lq with1 ď q ă p˚.

b) The embedding W 1,ppUq ãÑ Lp˚

pUq is never compact, see Example 6.12in [AF]. ♦

3.2. Density and embedding theorems 57

We conclude this section with an interpolation estimate for first deriva-tives. Again there are plenty of variants.

Proposition 3.31. Let 1 ď p ă 8. Let either f P W 2,p0 pUq or U have

a compact Lipschitz boundary and f P W 2,ppUq. Then there are constantsC, ε0 ą 0 such that

ˆ dÿ

j“1

‖Bjf‖pp˙1p

ď ε

ˆ dÿ

i,j“1

‖Bijf‖pp˙1p

`C

ε‖f‖p,

for all ε ą 0 if f PW 2,p0 pUq and for all 0 ă ε ď ε0 if f PW 2,ppUq.

Proof.12 1) Let f P C2c pUq and extend it to Rd by 0. Take j “ 1. Write

x “ pt, yq P R ˆ Rd´1 for x P Rd. Fix y P Rd´1 and set gptq “ fpt, yq fort P R. Let ε ą 0 and a, b P R with b ´ a “ ε. Take any r P pa, a ` ε

3q andt P pb´ ε

3 , bq. There there is a number s “ spr, tq P pa, bq such that

|g1psq| “ˇ

ˇ

ˇ

gptq ´ gprq

t´ r

ˇ

ˇ

ˇď

3

εp|gptq|` |gprq|q.

For every s P pa, bq we thus obtain

|g1psq| “ˇ

ˇ

ˇg1psq `

ż s

sg2pτq dτ

ˇ

ˇ

ˇď

3

εp|gprq|` |gptq|q `

ż b

a|g2pτq|dτ.

Integrating first over r and then over t, we conclude

ε

3|g1psq| ď 3

ε

ż a` ε3

a|gprq|dr ` |gptq|` ε

3

ż b

a|g2pτq|dτ,

ε2

9|g1psq| ď

ż a` ε3

a|gprq|dr `

ż b

b´ ε3

|gptq|dt` ε2

9

ż b

a|g2pτq|dτ,

|g1psq| ď 9

ε2

ż b

a|gpτq| dτ `

ż b

a|g2pτq| dτ

ď ε1p1

9

ε2

´

ż b

a|gpτq|p dτ

¯1p

` ε1p1

´

ż b

a|g2pτq|p dτ

¯1p

ď εp´1p 2

p´1p

ˆˆ

9

ε2

˙p ż b

a|gpτq|p dτ `

ż b

a|g2pτq| dτ

˙1p

,

where we used Holder’s inequality first for the integrals and then in R2. Wetake now the p-th power and then integrate over s arriving at

ż b

a|g1psq|p ds ď εεp´12p´1

ˆ

9p

ε2p

ż b

a|gpτq|p dτ `

ż b

a|g2pτq|p dτ

˙

.

Now choose a “ ak “ kε and b “ bk “ pk ` 1qε for k P Z. Summing theintegrals on rkε, pk ` 1qεq for k P Z and then integrating over y P Rd´1, itfollows that

ż

R|g1pτq|p dτ ď εp2p´1

ˆ

9p

ε2p

ż

R|gpτq|p dτ `

ż

R|g2pτq|p dτ

˙

,

ż

U|B1f |p dx ď p2εqp

ż

U|B11f |p dx`

36p

p2εqp

ż

U|f |p dx. (3.24)

12Not shown in the lectures.

3.3. Traces 58

2) By approximation, (3.24) can be established for all f P W 2,p0 pUq. The

same result holds for Bjf and Bjjf with j P t2, . . . , du. We now replace 2εby ε, sum over j and take the p-th root to arrive at

ˆ dÿ

j“1

‖Bjf‖pp˙1p

ď

ˆ

εpdÿ

j“1

‖Bjjf‖pp `36p

εp‖f‖pp

˙1p

ď ε

ˆ dÿ

j“1

‖Bjjf‖pp˙1p

`36

ε‖f‖p, (3.25)

for all f PW 2,p0 pUq, as asserted.

3) If u P W 2,ppUq and U has a Lipschitz boundary, we use the extensionoperator EU P BpW 2,ppUq,W 2,ppRdqq from Theorem 3.24 to deduce from(3.25) with U “ Rd that

ˆ dÿ

j“1

‖Bjf‖pLppUq

˙1p

ď

ˆ dÿ

j“1

‖BjEUf‖pLppRdq

˙1p

ď ε

ˆ dÿ

j“1

‖BjjEUf‖pLppRdq

˙1p

`36

ε‖EUf‖LppRdq

ď ε ‖EUf‖W 2,ppRdq `36

ε‖EUf‖LppUq

ď cε ‖f‖W 2,ppUq `c

ε‖f‖LppUq

ď c0ε

ˆ dÿ

i,j“1

‖Bijf‖pp˙1p

` c1ε

ˆ dÿ

j“1

‖Bjf‖pp˙1p

`c

ε‖f‖p

where we assume that ε P p0, 1s and the constants c, c0, c1 do not depend onε or f . Choosing ε1 “ mint 1

2c1, 1u we arrive at

1

2

ˆ dÿ

j“1

‖Bjf‖pp˙1p

ď c0ε

ˆ dÿ

i,j“1

‖Bijf‖pp˙1p

`c

ε‖f‖p

if 0 ă ε ď ε1. This inequality implies the assertion for U with a Lipschitzboundary, after replacing ε by εp2c0q and ε1 by ε0 “ mintc0c1, 2c0u.

Remark 3.32. Arguing as in part 1) of the above proof, one derivesProposition 3.31 on W 2,ppJq for every open interval J Ď R with uniformconstants. ♦

3.3. Traces

For p ą d and Lipschitz boundaries, we have W 1,ppUq ãÑ CpUq bySobolev’s Theorem 3.26 so that the trace map f ÞÑ f|BU is well de-

fined from W 1,ppUq to CpBUq, say. Theorem 3.11 further implies thatW 1,1pa, bq ãÑ Cpra, bsq for d “ 1. In other cases it is not clear at all how togive a meaning to the mapping f ÞÑ f |pBUq on W 1,ppUq since for reasonableopen sets BU is a null set. The next result solves this problem. Moreover,it says that W 1,p

0 pUq is the space of functions in W 1,ppUq with trace 0.

3.3. Traces 59

Theorem 3.33 (Trace theorem). Let p P r1,8q and U Ď Rd be open witha compact boundary BU of class C1. Then the trace map f ÞÑ f|BU from

W 1,ppUqXCpUq to LppBU, σq has a bounded linear extension tr : W 1,ppUq Ñ

LppBU, σq whose kernel is W 1,p0 pUq, where σ is the surface measure on BU .

Proof.13 1) Let u P C1pUq. By the definition of the surface integral,see Section 2.3 with a slightly different notation, there are finitely many

diffeomorphisms Ψj : rUj Ñ rVj and ϕj P C1c prUjq with 0 ď ϕj ď 1 such that

upLppBU,σq is dominated by

cmÿ

j“1

ż

Vj

ϕj ˝Ψ´1j |u ˝Ψ´1

j |p dy1

where rUj and rVj are open subsets of Rd, the sets rUj cover BU , the functions

ϕj form a partition of unity subordinated to rUj , Vj :“ tpy1, ydq P rVjˇ

ˇ yd “ 0u,

Vj` :“ tpy1, ydq P rVjˇ

ˇ yd ą 0u, ΨjprUj X BUq “ Vj , and ΨjprUj X Uq “ Vj`.

We set v “ u˝Ψ´1j and ψ “ ϕj ˝Ψ´1

j P C1c p

rVjq and drop the indices j below.By means of Fubini’s theorem and the fundamental theorem of calculus, wecomputeż

Vψ |vpy1q|p dy1 “ ´

ż

V`

Bdpψ |v|pq dy

“ ´

ż

V`

rpBdψq |v|p ` pψ|v|p´2 RepvBdvqsdy

ď c

ż

V`

r|v|p ` |v|p´1 |Bdv|s dy ď c vpp ` c vp´1p Bdvp

ď c pvpp ` Bdvppq ď c vp

W 1,ppV`qď c up

W 1,ppUq.

Here we also used Holder’s and Young’s inequality, Proposition 3.9 and thetransformation rule. As a result, the map tr : pC1pUq, ¨ 1,pq Ñ LppBU, σq,tru “ u|BU , is continuous. Theorem 3.27 allows to extend tr to an operatorin LpW 1,ppUq, LppBU, σqq. If we start with a function u P W 1,ppUq X CpUq,then we can construct approximations un P C1pUq which converge to uin W 1,ppUq and in CpUq, see the proof of Theorem 5.3.3 in [Ev]. Hence,trun “ un|BU tends to u|BU uniformly on BU and to tru in LppBU, σq, sothat tru “ u|BU .

2a) We next observe that the inclusion W 1,p0 pUq Ď Nptrq is a consequence

of the continuity of tr since tr vanishes on C8c pUq and this space is dense

in W 1,p0 pUq by definition. To prove the converse, we start with the model

case that v P W 1,ppV`q has a compact support in V` and tr v “ 0. Ourdensity results yield vn P C

1pV`q converging to v in W 1,ppV`q, and hencetr vn “ vn|V Ñ 0 in LppV q, as nÑ8. Observe that

|vnpy1, ydq| ď |vnpy

1, 0q| `

ż yd

0|Bdvnpy

1, sq|ds,

|vnpy1, ydq|

p ď 2 |vnpy1, 0q|p ` 2

ˆż yd

0|Bdvnpy

1, sq|ds

˙p

13Not shown in the lectures.

3.3. Traces 60

for y1 P V and yd ě 0. Integrating over y1 und employing Holder’s inequality,we obtainż

V|vnpy

1, ydq|p dy1 ď 2

ż

V|vnpy

1, 0q|p dy1 ` 2yp´1d

ż

V

ż yd

0|Bdvnpy

1, sq|p dsdy1.

We can now let nÑ8 and arrive atż

V|vpy1, ydq|

p dy1 ď 2yp´1d

ż

V

ż yd

0|Bdvpy

1, sq|p dsdy1. (3.26)

We next use a cutoff argument to obtain a support in the interior of V`.Choose a function χ P C8pR`q such that χ “ 0 on r0, 1s and χ “ 1 onr2,8q. Set χnpsq “ χpnsq for s ě 0 and n P N, and define wn “ χnv on V`.Note that wn Ñ v in LppV`q as nÑ8, Bjwn “ χnBjv for j P t1, . . . , d´ 1uand Bdwn “ χnBdv ` nχ

1pn¨qv. Estimate (3.26) further impliesż

V`

|∇wn ´∇v|pp dy

ď c

ż 2n

0

ż

V|1´ χn|

p |∇v|pp dy1 ds` cnpż 2n

0

ż

V|vpy1, sq|p dy1 ds

ď c

ż 2n

0

ż

V|∇v|pp dy1 ds` cnp

ż 2n

0sp´1

ż s

0

ż

V|Bdvpy

1, τq|p dy1 dτ ds

ď c

ż 2n

0

ż

V|∇v|pp dy1 ds` c

ż 2n

0

ż

V|Bdvpy

1, τq|p dy1 dτ

for some constants c ą 0. Because of v PW 1,ppV`q, the above integrals tendto 0 as n Ñ 8, and so wn Ñ v in W 1,ppV`q as n Ñ 8. Since wn “ 0 foryd P p0, 1ns, we can mollify wn to obtain a function pwn P C

8c pV`q such that

pwn ´ wn1,p ď 1n. This means that pwn Ñ v in W 1,ppV`q as nÑ8.

2b) We come back to u PW 1,ppUq and consider the sets rUj and rVj and the

functions Ψj and ϕj from step 1). Let vj “ pϕjuq ˝Ψ´1j . First, observe that

the trace of vj to the set Vj is given by ptrϕjq ˝Ψ´1j ptruq ˝Ψ´1

j if u P CpUq,

in addition. By continuity one can extend this identity to all u P W 1,ppUq.

Let tru “ 0. Then we can apply part 2a) to vj and obtain pwjn P C1c pVj`q

converging to vj in W 1,ppVj`q. The function

pun “mÿ

j“1

pwjn ˝ pΨj |U X rUjq

thus belongs to C1c pUq and converges to u in W 1,ppUq as n Ñ 8. Since

pun has compact support, we can mollify pun to a function un P C8c pUq with

pun ´ un1,p ď 1n. This means that un Ñ u in W 1,ppUq as n Ñ 8, and

hence u PW 1,p0 pUq.

Remark 3.34. As in Remark 3.25 we give a much shorter proof in thecase p “ 2 of an even stronger continuity property of tr, but here only inthe half space case Rd` “ tpx, yq

ˇ

ˇx P Rd´1, y ą 0u with d ě 2.

1) Let u P C1c pRd`q “ C1pRd`qXCcpRd`q and let Fx be the Fourier transform

with respect to x P Rd´1. The map y ÞÑ pFxuqpξ, yq then belongs to C1c pR`q

3.3. Traces 61

for each ξ P Rd´1. Since we deal with regular functions, we can calculateż

Rd´1

|ξ|2 |pFxuqpξ, 0q|2 dξ

“ ´

ż

Rd´1

|ξ|2

ż 8

0By |pFxuqpξ, yq|2 dy dξ

“ ´2 Re

ż 8

0

ż

Rd´1

|ξ|2 Fxu BypFxuqdξ dy

ď 2

„ż 8

0

ż

Rd´1

|ξFxu|22 dξ dy

12„ż 8

0

ż

Rd´1

|BypFxuq|2 dξ dy

12

“ 2

„ż 8

0

ż

Rd´1

|Fxp∇xuq|22 dξ dy

12„ż 8

0

ż

Rd´1

|FxpByuq|2 dξ dy

12

“ 2

„ż 8

0

ż

Rd´1

|∇xu|22 dx dy

12„ż 8

0

ż

Rd´1

|Byu|2 dx dy

12

ď

ż

Rd`|∇xu|

22 dz `

ż

Rd`|Byu|

22 dz ď u2

W 1,2pRd`q,

also using Holder’s inequality, (3.21) and Plancherel’s theorem. For s ą 0we define the Bessel potential space

HspRnq “ tv P L2pRnqˇ

ˇ |ξ|s2 pv P L2pRnqu

endowed with the (Hilbertian) norm given by

v2Hs “

ż

Rnp1` |ξ|2s2 q |pvpξq|

2 dξ.

(Recall from (3.20) that HkpRnq “ W k,2pRnq for k P N.) We have thus

shown that the trace map is continuous from pC1c pRd`q, ¨ 1,2q to H

12 pRd´1q.

2) Theorem 3.27 implies that W 1,2pRd`q X C1pRd`q is dense in W 1,2pRd`q.Using cut-off functions as in the first part of Theorem 3.15, one sees the

density of C1c pRd`q in W 1,2pRd`q X C1pRd`q for ¨ 1,2. Hence, the trace map

is continuous from W 1,2pRd`q to H12 pRd´1q.

3) It is possible to show that tr : W 1,2pRd`q Ñ H12 pRd´1q is surjective.

One can show analogous boundedness and surjectivity results on domains Uwith a compact C1–boundary, and also for p P p1,8q (employing somewhatdifferent spaces of functions on BU). See e.g. Theorem 7.39 in [AF]. ♦

We can now prove the divergence theorem in Sobolev spaces, which is acrucial tool in many applications to partial differential operators

Theorem 3.35 (Gauß and Green formulas). Let U “ Rd or U Ď Rd havea compact boundary BU of class C1 with the outer unit normal ν, p P r1,8s,

F PW 1,ppUqd, and ϕ PW 1,p1pUq. We obtain

ż

UdivpF qϕdx “ ´

ż

UF ¨∇ϕdx`

ż

BUϕν ¨ F dσ. (3.27)

3.3. Traces 62

If u PW 2,ppUq and v PW 2,p1pUq, we obtainż

Up∆uqv dx “ ´

ż

U∇u ¨∇v dx`

ż

BUpBνuqv dσ

“

ż

Uu∆v dx`

ż

BUppBνuqv ´ uBνvq dσ.

(3.28)

If U “ Rd, these formulas hold without the boundary integrals. (We omit tr

in the boundary integrals and set Bνu “řdj“1 νj tr Bju.)

Proof. We first observe that Green’s formulas (3.28) are a straightfor-ward consequence of (3.27) with F “ ∇u.

1) Let U be bounded. Gauß’ formula (3.27) is shown in analysis coursesfor F P C1pUqd and ϕ P C1pUq.

a) Let p P p1,8q. For F P W 1,ppUqd and ϕ P W 1,p1pUq, Theorem 3.27provides functions Fn P C

1pUqd and ϕn P C1pUq that converge to F and ϕ

in W 1,ppUqd and W 1,p1pUq as n Ñ 8, respectively, since p, p1 ă 8. The-orem 3.33 then yields that Fn|BU Ñ trF in LppU, σqd and ϕn|BU Ñ trϕ

in Lp1

pU, σq as n Ñ 8. Since Bj : W 1,qpUq Ñ LqpUq is continuous forq P tp, p1u, we further obtain that the terms with derivatives converge in Lp,

respectively in Lp1

. Formula (3.27) now follows by approximation.b) Next, let p “ 1 and thus p1 “ 8. As above, divFn and Fn converge

to divF and F in L1pUq and L1pUqd, respectively, as well as Fn|BU Ñ trFin L1pBU, σqd. By Proposition 3.28, the function ϕ belongs to CpUq andone can thus extend it a map ϕ P CcpRdq. Set ϕn “ G 1

nϕ P C8c pRdq.

Properties (3.7) and (3.6) and Lemma 3.4 imply that ϕn Ñ ϕ in CpUq,∇ϕn8 ď ∇ϕ8 and ∇ϕn Ñ ∇ϕ pointwise a.e., as n Ñ 8, where wepossibly pass to a subsequence also denoted by pϕnqn. We can now take thelimit nÑ8 in the first and the third integral of equation (3.27) for Fn andϕn. For the second integral we also use the estimateˇ

ˇ

ˇ

ż

UFn ¨∇ϕn dx´

ż

UF ¨∇ϕdx

ˇ

ˇ

ˇď Fn´F 1 ∇ϕn8`

ż

U|F | |∇ϕn´∇ϕ| dx

and Lebesgue’s theorem for the integral on the right-hand side. The casep “ 8 is treated analogously.

2) Let U be unbounded. There is a radius r ą 0 such that BU Ď Bp0, rq.For k P N with k ą r we define cut-off functions χk as the functions ϕn inpart 1) of the proof of Theorem 3.15. For Fk “ χkF and ϕk “ χkϕ, theformula (3.27) follows from step 1) replacing U by U X Bp0, 2kq. Observethat Fk “ F and ϕk “ ϕ on BU . We compute

ż

UdivpFkqϕk dx “

ż

Uχ2k divpF qϕdx`

ż

UχkϕF ¨∇χk dx,

ż

UFk ¨∇ϕk dx “

ż

Uχ2kF ¨∇ϕdx`

ż

UχkϕF ¨∇χk dx.

The products ϕdivpF q, ϕF and F ¨∇ϕ are integrable, 0 ď χk ď 1, χk Ñ 1

pointwise and χ1k8 ď ck. Letting k Ñ 8, the theorem of dominatedconvergence implies (3.27) for F and ϕ. Note that this proof also works forU “ Rd, where BU “ H.

3.4. Differential operators in Lp spaces 63

3.4. Differential operators in Lp spaces

We now use some of above results to study differential operators in Lp

spaces and discuss their closedness, spectra and the compactness of theirresolvents.

Example 3.36. Let U Ď Rd be open, 1 ď p ă 8, X “ LppUq, and α P Nd0.Then the operator A “ Bα with DpAq “ tu P Dα,ppUq

ˇ

ˇu, Bαu P Xu is closed.

Let J Ď R be an open interval and X “ LppJq. Then also Ak “ Bk with

DpAkq “W k,ppJq for k “ 1 or k “ 2 are closed.Proof. Let un P DpAq with un Ñ u and Aun “ Bαun Ñ v in X as

n Ñ 8. By Lemma 3.5(b), the function u P X belongs to Dα,ppUq andBαu “ v P X. Hence, u P DpAq and Au “ v; i.e., A is closed.

In particular, for J “ U the operators A1 and B2 on D2 “ tu P

D2,ppJqˇ

ˇu, B2u P LppJqu are closed. Since DpA2q Ď D2, it remains toshow the converse inclusion. Let u P D2. Lemma 3.5(a) yields functionsun P C

8pJq such that un Ñ u and u2n Ñ B2u in LplocpJq, as n Ñ 8. FromProposition 3.31 and Remark 3.32 we obtain a constant c ą 0 such that forall compact intervals J 1 Ď J and g P C2pJq we have

g1LppJ 1q ď c pgLppJ 1q ` g2LppJ 1qq.

If we insert here un ´ um “ g, we see that u1n converges in LplocpJq to afunction v and hence u belongs to D1,ppJq and Bu “ v by Lemma 3.5(b).We can now conclude that

BuLppJ 1q “ limnÑ8

BunLppJ 1q ď c limnÑ8

punLppJ 1q ` B2unLppJ 1qq

“ cpuLppJ 1q ` B2uLppJ 1qq ď cpuLppJq ` B

2uLppJqq.

As a result, Bu P X and u P DpA2q. l

We note that in the one-dimensional case Bk is closed on W k,ppJq for allk P N (by similar proofs based on higher order versions of Proposition 3.31).This property relies on the fact that the domain W k,ppJq does not containderivatives in other directions. In contrast, B1 on W 1,ppp0, 1q2q is not closedin Lppp0, 1q2q which can be shown by functions unpx, yq “ ϕnpyq whereϕn P C

1pr0, 1sq converges in Lpp0, 1q to a function ϕ RW 1,pp0, 1q.Next, in various settings we study the first derivative in more detail.

Example 3.37. Let 1 ď p ă 8, X “ LppRq and A “ B with DpAq “W 1,ppRq. Then σpAq “ iR, σppAq “ H and

pRpλ,Aqfqptq “

#

ş8

t eλpt´sqfpsqds, Reλ ą 0,

´şt´8

eλpt´sqfpsq ds, Reλ ă 0,

for t P R and f P X, cf. Example 1.21 for X “ C0pRq.Proof. 1) Let Reλ ą 0 and ϕλ “ eReλ1R´ . By Rλfptq we denote

the integral in the assertion. Since |Rλpfptqq| ď pϕλ ˚ |f |qptq for all t P R,Young’s inequality (see e.g. Theorem 2.14 in [Sc2]) yields that

‖Rλf‖p ď ‖ϕλ‖1 ‖f‖p “1

Reλ‖f‖p.

Let fn P C8c pRq converge to f in LppRq. We then compute ddtRλfn “

λRλfn ´ fn, which tends to λRλf ´ f in LppRq by the above estimate.

3.4. Differential operators in Lp spaces 64

Since A is closed by Example 3.36, the function Rλf belongs to DpAq andARλf “ λRλf ´ f ; i.e., λI ´A is surjective.

2) Let Au “ µu for some u P DpAq and µ P C. Theorem 3.11 says that uis continuous so that Remark 3.12c) implies the continuous differentiabilityof u and so u1 “ µu. Consequently, u is equal to a multiple of eµ. Sinceeµ R X, we obtain u “ 0 and hence µI ´A is injective. We have shown thatσppAq “ H and λ P ρpAq with Rpλ,Aq “ Rλ if Reλ ą 0. In the same way,one sees that λ P ρpAq if Reλ ă 0, with the asserted resolvent.

3) Let λ P iR. Take ϕn P C1c pRq with ϕn “ n´1p on r´n, ns, ϕnptq “ 0

for |t| ě n ` 1 and ‖ϕn‖8 ď 2n´1p for every n P N. We then deriveun “ ϕneλ PW

1,ppRq, Aun “ λun ` ϕneλ,

‖un‖p ěˆż n

´n|ϕnptqeλptq|p dt

˙1p

“ n´1pp2nq

1p “ 21p,

‖λun ´Aun‖p “ ‖ϕ1neλ‖p “´

ż

tnď|t|ďn`1u|ϕ1nptqeλptq|p dt

¯1p

ď 2n´1p2

1p,

for every n P N. Consequently, λ P σappAq and so σpAq “ iR. l

Example 3.38. Let 1 ď p ă 8.a) Let X “ Lpp0, 1q and A “ B with DpAq “ W 1,pp0, 1q. Then σpAq “

σppAq “ C since eλ P DpAq and Aeλ “ λeλ for all λ P C.

b) Let X “ Lpp0, 1q and A “ B with DpAq “ tu P W 1,pp0, 1qˇ

ˇup0q “ 0u

(using the continuous representative of u P W 1,pp0, 1q from Theorem 3.11).

Then A is closed, σpAq “ H, Rpλ,Aqfptq “ ´şt0 eλpt´sqfpsqds for t P p0, 1q,

f P X and λ P C, and A has compact resolvent.Proof. As in Example 3.37 one sees that the above integral defines a

bounded inverse of λI ´ A for all λ P C. In particular, A is closed by Re-mark 1.11. (Alternatively, take un in DpAq such that un Ñ u and Aun Ñ vin X. Example 3.36 yields that u P W 1,pp0, 1q and Bu “ v. Using Theo-rem 3.11, we further infer that 0 “ unp0q Ñ up0q. Hence, u P DpAq andAu “ Bu “ v.) Finally, Remark 2.14 and Theorem 3.29 imply that A has acompact resolvent. l

c) Let X “ Lpp0,8q and A “ B with DpAq “ W 1,pp0,8q. ThenσpAq “ tλ P C

ˇ

ˇ Reλ ď 0u, σppAq “ tλ P Cˇ

ˇ Reλ ă 0u and Rpλ,Aqfptq “ş8

t eλpt´sqfpsqds for t ą 0, f P X and Reλ ą 0.Proof. The operator is closed by Example 3.36. As in Example 3.37,

one computes the resolvent for Reλ ą 0 and checks that iR does not containeigenvalues. If Reλ ă 0, then eλ is an eigenfunction. Since σpAq is closed,it is equal to tλ P C

ˇ

ˇ Reλ ă 0u. l

d) Let X “ Lpp0,8q and A “ ´B with DpAq “ W 1,p0 p0,8q. Then A

is closed, σpAq “ tλ P Cˇ

ˇ Reλ ď 0u, σppAq “ H, and Rpλ,Aqfptq “şt0 e´λpt´sqfpsqds for t ą 0, f P X and Reλ ą 0.Proof. If Reλ ą 0, the formula for the resolvent is verified as in Exam-

ple 3.37, so that A is closed. The point spectrum is empty since the onlypossible eigenfunctions eλ do not fulfill the boundary condition in DpAq.

Let Reλ ă 0 and f “ 1p0,1q. If u P DpAq satisfies λu ` Au “ f , thenup0q “ 0 and Bu “ ´λu´f is continuous except at t “ 1. By Remark 3.12c),

3.4. Differential operators in Lp spaces 65

u is piecewise C1 and hence

uptq “ e´λtż 1

0eλs ds, t ě 1.

So u cannot belong to X and thus f is not in the range of λI ´ A; i.e.,λ P σpAq. The result then follows from the closedness of the spectrum. l

e) Let X “ Lpp0, 1q and A “ B with DpAq “ tu PW 1,pp0, 1qˇ

ˇup0q “ up1qu.Then A is closed, σpAq “ σppAq “ 2πiZ and A has a compact resolvent.(These facts can be proved as in Example 2.17 for X “ Cpr0, 1sq, using nowTheorem 3.29 for the compactness.) ♦

We now turn our attention to the second derivative and the Laplacian.

Example 3.39. Let X “ LppRq, 1 ď p ă 8, and A “ B2 with DpAq “W 2,ppRq. Then σpAq “ R´.Proof. 1) Set A1 “ B and DpA1q “ W 1,ppRq. Let µ P CzR´. There

exists a number λ P C with Reλ ą 0 such that µ “ λ2. Observe that

pµI ´Aqu “ pλI ´A1qpλI `A1qu (3.29)

for all u P DpAq. Example 3.37 says that λI ˘ A1 are invertible. Hence,µI ´A is injective. Next, for v PW 1,ppRq the function

BpλI `A1q´1v “ A1pλI `A1q

´1v “ ´λpλI `A1q´1v ` v

belongs to W 1,ppRq. This means that pλI`A1q´1 maps W 1,ppRq into DpAq.

Given f P X, the map u :“ pλI `A1q´1pλI ´A1q

´1f thus is an element ofDpAq and µu´Au “ f in view of the factorization (3.29) . We have shownthat µ P ρpAq and Rpµ,Aq “ pλI `A1q

´1pλI ´A1q´1.

2) For µ ď 0, we have µ “ λ2 for some λ P iR and (3.29) is still true. Theoperator λI´A1 is not surjective since its range is not closed by Example 3.37and Proposition 1.19. Equation (3.29) thus implies that µI ´ A is notsurjective. As a result, σpAq “ R´. l

Example 3.40. Let X “ Lpp0, 1q, 1 ď p ă 8, and A “ B2 with DpAq “

W 2,pp0, 1q XW 1,p0 p0, 1q. Then A is closed, σpAq “ σppAq “ t´π

2k2ˇ

ˇ k P Nu,and A has a compact resolvent. These facts can be proved as in Example 2.17for X “ Cpr0, 1sq, using now Theorem 3.29 for the compactness. ♦

Example 3.41. Let X “ L2pRdq and A “ ∆ “ B11 ` . . . ` Bdd withDpAq “W 2,2pRdq. Then A is closed and σpAq “ R´.Proof. We use the Fourier transform F which is unitary on X, see e.g.

Theorem 5.11 in [Sc1]. By (3.20) and (3.21), we have

∆u “ F´1p´|ξ|22puq for u P DpAq “ tu P X | |ξ|22 pu P Xu.

1) Let λ P CzR´. Set mλpξq “ pλ ` |ξ|22q´1 for ξ P Rd. Observe that

mλ8 ď cλ where cλ “ 1| Imλ| if Reλ ď 0 and cλ “ 1|λ| if Reλ ą 0. Let

f P X. Then mλpf belongs to X so that Rλf :“ F´1pmλ

pfq P X and

Rλf2 “ mλpf2 ď cλ pf2 “ cλ f2 .

Moreover, since |ξ|22mλpξq is bounded for ξ P Rd, the function |ξ|22FRλf “|ξ|22mλ

pf is an element of X; i.e., Rλf P DpAq. Similarly we see that

pλI ´∆qRλf “ F´1pλ` |ξ|22qFF´1mλpf “ f,

3.4. Differential operators in Lp spaces 66

RλpλI ´∆qu “ F´1mλFF´1pλ` |ξ|22qpu “ u, u P DpAq.

Thus, λ P ρpAq and Rpλ,Aq “ Rλ. In particular, A is closed and σpAq Ď R´.2) Let λ ă 0. Define mλpξq “ pλ ` |ξ|

22q´1 for ξ P Rd with |ξ| ‰ ´λ and

mλpξq “ 0 if |ξ| “ ´λ. Then mλ is unbounded. Suppose that mg P L2pRdqfor all g P L2pRdq. Then the (closed) multiplication operator M : g ÞÑ mg inL2pRdq was defined on L2pRdq and thus bounded by the closed graph theo-rem. Set Bn “ tξ P Rd

ˇ

ˇ |mλpξq| ě nu for n P N. Observe that the Lebesgue

measure `n of Bn is contained in p0,8q. Set gn “ `´12n mλ |mλ|

´11Bn .

Since |gn| “ `´12n 1Bn and |Mgn| “ `

´12n |mλ|1Bn , we obtain gn2 “ 1 and

Mgn2 ě n for all n P N. These relations contradict the continuity of M .

So there exists a function h P X such that mλph R X. If there was an

element u of DpAq with λu ´ ∆u “ h, we would obtain as above that

λpu ` |ξ|22pu “ph and hence mλ

ph “ pu P X which is impossible. As a result,λ P σpAq and σpAq “ R´. l

Example 3.42. Let 1 ă p ă 8, X “ LppRdq, and A “ ∆ with DpAq “W 2,ppRdq. Then A is closed.Proof. The Calderon–Zygmund estimate says that the graph norm of

A is equivalent to ‖¨‖2,p on C8c pRdq, see e.g. Corollary 9.10 in [GT]. Let

u PW 2,ppRdq. By Theorem 3.15, there are un P C8c pRdq that converge to u

in W 2,ppRdq as nÑ8, and hence un Ñ u and ∆un Ñ ∆u in X. We derive

u2,p “ limnÑ8

un2,p ď limnÑ8

c punp ` ∆unpq

“ c pup ` ∆upq ď c1 u2,p ,

so that ¨ A is equivalent to a complete one and thus A is closed. l

Example 3.43 (Dirichlet Laplacian). Let 1 ă p ă 8, U Ď Rd be boundedand open with BU P C2, X “ LppUq, and A “ ∆ with DpAq “ W 2,ppUq X

W 1,p0 pUq. Then A is closed, invertible and has a compact resolvent. In

particular, σpAq “ σppAq.Proof. The closedness of A follows from Theorem 9.14 in [GT]. Its

bijectivity is shown in Theorem 9.15 of [GT]. Remark 2.14, Theorem 3.29and Theorem 2.16 then imply the other assertions. l

There are variants of Examples 3.42 and 3.43 for X “ L1pUq, X “ L8pUqand in other sup-norm spaces, see Theorem 5.8 in [Ta] as well as Sec-tions 3.1.2 and 3.1.5 in [Lu]. In theses cases, the descriptions of the domainsare much more complicated and they are not just (subspaces of) Sobolev(or C2-) spaces. To indicate the difficulties, we note that there is a functionu RW 2,8pBp0, 1qq such that ∆u P L8pBp0, 1qq, namely

upx, yq “

#

px2 ´ y2q lnpx2 ` y2q for px, yq ‰ p0, 0q,

0 for px, yq “ p0, 0q.

Then the second derivative

Bxxupx, yq “ 2 lnpx2 ` y2q `4x2

x2 ` y2`p6x2 ´ 2y2qpx2 ` y2q ´ 4x2px2 ´ y2q

px2 ` y2q2

is unbounded around p0, 0q, but ∆upx, yq “ 8x2´y2

x2`y2is bounded.

CHAPTER 4

Selfadjoint operators

In this chapter X and Y are Hilbert spaces with scalar product p¨|¨q ,unless something else is said.

4.1. Basic properties

Let A be a densely defined linear operator from X to Y . We define theHilbert space adjoint A1 as in the Banach space case by

DpA1q :“ ty P Yˇ

ˇ D z P X @x P DpAq : pAx|yq “ px|zqu,

A1y :“ z. (4.1)

As in Remark 1.23 one sees that A1 : DpA1q Ñ X is a linear map, which isclosed from Y to X. Let T P BpX,Y q. Then DpT 1q “ Y and T 1 is given by

@ x P X, y P Y :`

xˇ

ˇT 1y˘

“ pTx|yq ,

as in (5.9) in [Sc2]. We recall from Proposition 5.42 of [Sc2] that

T “ T 1, T 2 “ T, pαT ` βSq1 “ αT 1 ` βS1, pUT q1 “ T 1U 1

for T, S P BpX,Y q, U P BpY, Zq, a Hilbert space Z, and α, β P C.

Definition 4.1. A densely defined linear operator A on X is called self-adjoint if A “ A1 (in particular, DpAq “ DpA1q and A must be closed) skew-adjoint if A “ ´A1, and normal if AA1 “ A1A. We say that T P BpX,Y q isunitary if T is invertible with T´1 “ T 1.

Let Φ : X Ñ X˚ be the (antilinear) Riesz isomorphism given bypΦpxqqpyq “ py|xq for all x, y P X. For λ P C and X “ Y , we obtain

λIX ´ T1 “ Φ´1pλIX˚ ´ T

˚qΦ, λIX ´A1 “ Φ´1pλIX˚ ´A

˚qΦ,

with DpA1q “ Φ´1 DpA˚q, see p.109 of [Sc2]. Theorem 1.24 thus implies

σpAq “ σpA˚q “ σpA1q, σrpAq “ σppA˚q “ σppA

1q, (4.2)

Rpλ,A1q “ Φ´1Rpλ,A˚qΦ “ Φ´1Rpλ,Aq˚Φ “ Rpλ,Aq1 for λ P ρpAq,

where the bars mean complex conjugation of each element.

Remark 4.2. a) Selfadjoint, skewadjoint or unitary operators (with X “

Y ) are normal. If A “ A1 and λ P C, then λI ´A is normal.

b) A densely defined linear operator A is skewadjoint if and only if iA isselfadjoint. (Use that (4.1) yields piAq1 “ ´iA.) ♦

Theorem 1.16 says that the spectral radius rpT q is less or equal than T for each bounded operator T on a Banach space. Already for the matrixT “ p 0 1

0 0 q one has the strict inequality rpT q “ 0 ă 1 “ T . We next show‖T‖ “ rpT q for normal operators which is the key to their deeper properties.

67

4.1. Basic properties 68

Proposition 4.3. Let X and Y be Hilbert spaces. If T P BpX,Y q, then‖T 1T‖ “ ‖TT 1‖ “ ‖T‖2. Let T P BpXq be normal. We then have ‖T‖ “rpT q, and thus T “ 0 if σpT q “ t0u.

Proof. For x P X we compute

‖Tx‖2 “ pTx|Txq “`

T 1Tx|xq ď ‖T 1T‖‖x‖2,

‖T‖2 “ sup‖x‖ď1

‖Tx‖2 ď ‖T 1T‖ ď ‖T 1‖‖T‖ “ ‖T‖2;

i.e., ‖T‖2 “ ‖T 1T‖. We infer that T 2 “ T 12 “ T 2T 1 “ TT 1.Next, let T be normal. From the first part we then deduce

‖T 2‖2 “ ‖T 2pT 2q1‖ “ ‖TTT 1T 1‖ “ TT 1TT 1 “ ‖TT 1pTT 1q1‖“ ‖TT 1‖2 “ ‖T‖4,

so that ‖T 2‖ “ ‖T‖2. Iteratively it follows that ‖T 2n‖ “ ‖T‖2n for all n P N.Using Theorem 1.16, we conclude

rpT q “ limmÑ8

‖Tm‖1m “ limnÑ8

‖T 2n‖2´n “ ‖T‖.

The following concepts turn out to be very useful to compute adjoints,for instance.

Definition 4.4. Let A and B be linear operators from a Banach spaceX to a Banach space Y . We say that B extends A (and write A Ď B) ifDpAq Ď DpBq and Ax “ Bx for all x P DpAq.

Next, let X be a Hilbert space. A linear operator A on X is called sym-metric if we have pAx|yq “ px|Ayq for all x, y P DpAq.

Of course, a selfadjoint operator is symmetric. As we see in Example 4.8,the converse is not true in general.

Remark 4.5. Let A and B be linear operators from a Banach space Xto a Banach space Y .

a) The operator B extends A if and only if its graph GpBq contains GpAq.Let A Ď B. Then A “ B is equivalent to DpBq Ď DpAq.

b) Let A Ď B, A be surjective and B be injective. Then A and B areequal. Hence, if X “ Y and there is a λ P C such that λI ´ A is surjectiveand λI ´B is injective, then A “ B.Proof. Let x P DpBq and set y “ Bx. The surjectivity of A yields a

vector z P DpAq Ď DpBq such that y “ Az “ Bz. Since B is injective, weobtain x “ z P DpAq so that A “ B. l

c) Let A be densely defined and symmetric on a Hilbert space X. Defini-tion (4.1) implies that A Ď A1. In particular, A is selfadjoint if and only ifDpA1q Ď DpAq. ♦

In the next theorem we give spectral conditions implying that a symmetricoperator is selfadjoint. We start with a simple, but crucial lemma.

Lemma 4.6. Let A be symmetric, x P DpAq and α, β P R. Set λ “ α` iβ.We have pAx|xq P R and

‖λx´Ax‖2 “ ‖αx´Ax‖2 ` |β|2 ‖x‖2 ě |β|2 ‖x‖2.

If A is also closed, then σappAqĎR and ‖Rpλ,Aq‖ ď 1|Imλ| for all λPρpAqzR.

4.1. Basic properties 69

Proof. For x P DpAq we have pAx|xq “ px|Axq “ pAx|xq so thatpAx|xq “ px|Axq is real. From this fact we deduce that

‖λx´Ax‖2 “ pαx´Ax` iβx|αx´Ax` iβxq

“ ‖αx´Ax‖2 ` 2 Re piβx|αx´Axq ` ‖iβx‖2

“ ‖αx´Ax‖2 ` 2 Repiβα‖x‖2 ´ iβ px|Axqq ` |β|2 ‖x‖2

“ ‖αx´Ax‖2 ` |β|2 ‖x‖2 ě |β|2 ‖x‖2.

In particular, λ R σappAq if Imλ “ β ‰ 0.If λ P ρpAqzR and y P X, write x “ Rpλ,Aqy P DpAq. We then calculate

‖y‖2 “ ‖λx´Ax‖2 ě |Imλ|2 ‖x‖2 “ |Imλ|2 ‖Rpλ,Aqy‖2.

Theorem 4.7. Let X be a Hilbert space and A be densely defined, closedand symmetric.

(a) Then σpAq is either a subset of R or σpAq “ C or σpAq “ tλ PCˇ

ˇ Imλ ě 0u or σpAq “ tλ P Cˇ

ˇ Imλ ď 0u.(b) The following assertions are equivalent.

(i) A “ A1.(ii) σpAq Ď R.

(iii) iI ´A1 and iI `A1 are injective.(iv) piI ´AqDpAq and piI `AqDpAq are dense.

(c) If ρpAq X R ‰ H, then A is selfadjoint.(d) Let A be selfadjoint. Then we have

‖Rpλ,Aq‖ ď 1

|Imλ|(4.3)

for λ R R. Further, σpAq “ σappAq is non-empty and A has no selfadjointextension B ‰ A.

Proof. (a) Let λ P σpAq and µ P ρpAq. Suppose that Imλ ą 0 andImµ ą 0. The line segment from λ to µ then must contain a point γ P BσpAq.This point belongs to σappAq by Proposition 1.19 and satisfies Im γ ą 0,which contradicts Lemma 4.6 since A is symmetric. Similarly we excludethat Imλ ă 0 and Imµ ă 0. Since the spectrum is closed, only the fourcases in (a) remain.

(b) Let A be selfadjoint. Lemma 4.6 yields σappAq Ď R. Due to (4.2)we also have σrpAq “ σppA

1q “ σppAq. Hence, σrpAq “ σppAq Ď R. FromProposition 1.19 we thus deduce σpAq Ď R; i.e., (i) implies (ii). The im-plication ‘(ii)ñ(iii)’ is obvious. Equation (4.2) also shows that ˘i P σppA

1q

if and only if ¯i P σrpAq so that (iii) and (iv) are equivalent. Finally, let(iv) (and thus (iii)) be true. The range of the operator iI ˘ A is closed byLemma 4.6 and Proposition 1.19. In view of (iv), the map iI ˘ A is thensurjective. On the other hand, iI˘A1 is injective because of (iii), and henceA “ A1 thanks to Remark 4.5b).

(c) If there is a point λ in ρpAq X R, then ρpAq contains a ball around λby its openness. By part (a), the spectrum of A is thus contained in R, andso A is selfadjoint by (b).

(d) Let A “ A1. If its spectrum was empty, then A is invertible with aselfadjoint inverse. Proposition 1.20 thus yields that σpA´1q is equal to t0u

4.1. Basic properties 70

so that A´1 “ 0 by Proposition 4.3, which is impossible. Hence, σpAq ‰ H.Since σpAq “ σpA1q Ď R by (b), the other parts of assertion (c) follow fromLemma 4.6 and Remark 4.5.

Example 4.8. a) Let X “ L2pRq and A “ iB with DpAq “ W 1,2pRq.Then A is selfadjoint with σpAq “ R.Proof. For u, v P DpAq, integrating by parts we deduce

pAu|vq “ i

ż

RpBuqv ds “ ´i

ż

RuBv ds “

ż

RuiBv ds “ pu|Avq ,

see Theorem 3.35; i.e., A is symmetric. Example 3.37 and Proposition 1.20further imply that σpAq “ iσp´iAq “ i2 R “ R. Hence, A is selfadjoint. l

b) Let X “ L2p0,8q and A “ iB on DpAq “ W 1,2p0,8q. Then A is notsymmetric.Proof. For u, v P DpAq with up0q “ vp0q “ 1, as above an integration

by parts implies

pAu|vq “ i

ż 8

0pBuqv ds “ ´i

ż 8

0uBv ds´ iup0qvp0q “ pu|Avq ´ i. l

c) Let X “ L2p0,8q and A “ iB on DpAq “ W 1,20 p0,8q. Then A is

symmetric, but not selfadjoint, and σpAq “ tλ P Cˇ

ˇ Imλ ě 0u.Proof. Symmetry is shown as in a) using Theorem 3.35 and that now

u, v P DpAq have trace 0 at s “ 0. From Example 3.38 and Proposition 1.20we deduce that σpAq “ ´iσpiAq “ ´itλ P C

ˇ

ˇ Reλ ď 0u “ tλ P Cˇ

ˇ Imλ ě0u. Consequently, A is not selfadjoint. l

d) Let X “ L2pRdq and A “ ∆ with DpAq “ W 2,2pRdq. Then A isselfadjoint with σpAq “ R´.

Proof. For u, v P DpAq, Theorem 3.35 yields

pAu|vq “

ż

Rdp∆uqv dx “

ż

Rdu∆v dx “ pu|Avq ,

so that A is symmetric. In Example 3.41 we have seen that σpAq “ R´, andhence A is self-adjoint. l

e) Let U Ď Rd be open with C2–boundary and A “ ∆ with DpAq “

W 2,2pUq XW 1,20 pUq. Then A is selfadjoint (and has compact resolvent by

Example 3.43). In fact, the symmetry of A can be shown as in part d)because the traces of u, v P DpAq vanish. Then A is selfadjoint since it isinvertible by Example 3.43.

f) Let X “ L2p0, 1q, A0 “ B2 with DpA0q “ W 2,20 p0, 1q and A be as in

e) with U “ p0, 1q. As in e) we see that A0 is symmetric. But A0 is notselfadjoint, since A0 Ř A and A “ A1. We further claim that A10 “ B

2 withDpA10q “W 2,2p0, 1q.Proof. For v PW 2,2p0, 1q and u P DpAq, we deduce from Theorem 3.35

pA0u|vq “

ż 1

0pB2uqv ds “

ż 1

0uB2v ds` rvBu´ uBvs10 “ pu|B

2vq;

4.1. Basic properties 71

i.e., pB2,W 2,2p0, 1qq Ď A10. Conversely, take v P DpA10q. For u P C8c p0, 1q wehave u P C8c p0, 1q Ď DpAq and hence obtain

ż 1

0pB2uqv ds “ pA0u|vq “ pu|A

10vq “

ż 1

0uA10v ds.

After complex conjugation, we see that v P D2,2p0, 1q XX and B2v “ A10v PX. The function v thus belongs to W 2,2p0, 1q by Example 3.36. l

Theorem 4.9. Let A be densely defined and selfadjoint on the Hilbertspace X and let B be symmetric with DpAq Ď DpBq. Assume there areconstants c ą 0 and δ P r0, 12q such that ‖Bx‖ ď c‖x‖ ` δ‖Ax‖ for allx P DpAq. Then the operator A`B with DpA`Bq “ DpAq is selfadjoint.

Proof. By Theorem 4.7, the number it belongs to ρpAq for all t P Rzt0u.Take ε P p0, 1´ 2δq Ď p0, 1q and x P X. Using (4.3), we estimate

‖BpitI ´Aq´1x‖ ď δ‖ApitI ´Aq´1x‖` c ‖pitI ´Aq´1x‖“ δ ‖itpitI ´Aq´1x´ x‖` c ‖pitI ´Aq´1x‖

ď δ´ |t||t|` 1

¯

‖x‖` c

|t|‖x‖ ď p1´ εq‖x‖,

whenever |t| ě c1´2δ´ε . Theorem 1.27 now implies that ˘it P ρpA ` Bq for

such t. Moreover, A`B is symmetric since

ppA`Bqx|yq “ pAx|yq ` pBx|yq “ px|Ayq ` px|Byq “ px|pA`Bqyq

for all x, y P DpAq. So, A`B is selfadjoint due to Theorem 4.7.

Actually, in the above theorem it suffices to assume that δ ă 1, see The-orem X.13 in [RS].

Example 4.10. Let X “ L2pR3q and A “ ∆ with DpAq “W 2,2pR3q. SetV upxq “ b

|x|2upxq for x P R3, u P DpAq and some b P R. Then A ` V with

domain DpAq is selfadjoint.Proof. Since kp “ 4 ą d “ 3, we have DpAq ãÑ C0pR3q by Theorem 3.17.

Let 0 ă ε ď 1. Using polar coordinates and Example 3.41, we computeż

R3

|V u|2 dx “ b2ż

Bp0,εq

|upxq|2

|x|22dx` b2

ż

R3zBp0,εq

|upxq|2

|x|22dx

ď c ‖u‖28

ż ε

0

r2

r2dr `

b2

ε2

ż

R3zBp0,εq|u|2 dx

ď cε‖u‖22,2 `

b2

ε2‖u‖2

2 ď cε‖Au‖22 ` cε‖u‖2

2 `b2

ε2‖u‖2

2,

for constants c ą 0 independent of u P DpAq and ε. Moreover, V is sym-metric on DpAq since

pV u|vq “

ż

R3

b

|x|2upxqvpxq dx “ pu|V vq

for all u, v P DpAq. For small ε ą 0, Theorem 4.9 implies that A ` V isselfadjoint. l

4.2. The spectral theorems 72

The spectra σpA`V q and σppA`V q and the eigenfunctions of A`V arecomputed in §7.3.4 of [Tr], where b ą 0. The above operator A “ ∆` V isused in physics to describe the hydrogen atom, see also Example 4.19.

4.2. The spectral theorems

Hermitian matrices are unitarily equivalent to diagonal matrices and thusvery easy to treat. In this section we establish the infinite dimensionalanalogues of this basic result from linear algebra. The following results canbe extended to normal operators, and the separability assumption madebelow can be removed. See e.g. Corollaries X.2.8 and X.5.4 in [DS2] orTheorems 13.24, 13.30 and 13.33 in [Ru2].

Let T P BpZq for a Banach space Z and ppzq “ a0 ` a1z ` . . .` anzn be

a complex polynomial. We then define the operator polynomial

ppT q “ a0I ` a1T ` . . .` anTn P BpZq. (4.4)

This gives a map p ÞÑ ppT q from the space of polynomials to BpZq. Forselfadjoint T on a Hilbert space one can extend this map to all f P CpσpT qq,as seen in the next theorem. We set p1pzq “ z.

Theorem 4.11 (Continuous functional calculus). Let T P BpXq be selfad-joint on a Hilbert space X. There is exactly one map ΦT : CpσpT qq Ñ BpXq;f ÞÑ fpT q satisfying

(C1) pαf ` βgqpT q “ αfpT q ` βgpT q,(C2) ‖fpT q‖ “ ‖f‖8 (hence, ΦT is injective),(C3) 1pT q “ I and p1pT q “ T ,(C4) pfgqpT q “ fpT qgpT q “ gpT qfpT q,(C5) fpT q1 “ fpT q

for all f, g P CpσpT qq and α, β P C. In particular, we have ΦT ppq “ ppT qfor each polynomial p, where ppT q is given by (4.4).

Proof. We first show the properties (C1)-(C5) for polynomials pptq “a0` a1t` . . .` ant

n and qptq “ b0` b1t` . . .` bntn with t P R and the map

p ÞÑ ppT q defined by (4.4), where any aj , bj P C may be equal to 0. Clearly,(C1) and (C3) are true in this case, and ppT q1 “

řnj“0 ajpT

jq1 “ ppT q since

T “ T 1. Moreover,

ppqqpT q “2nÿ

m“0

´

ÿ

0ďj,kďnj`k“m

ajbk

¯

Tm “nÿ

j“0

ajTj

nÿ

k“0

bjTk “ ppT qqpT q

and so ppqqpT q “ pqpqpT q “ qpT qppT q; i.e., (C4) is shown for polynomials.Properties (C4) and (C5) imply that ppT q is normal. Hence, Proposition 4.3and Lemma 4.12 below imply

‖ppT q‖ “ maxt|λ|ˇ

ˇλ P σpppT qqu “ maxt|λ|ˇ

ˇλ P ppσpT qqu “ ‖p‖8.Let f P CpσpT qq. Since σpT q Ď R is compact, Weierstraß’ approximationtheorem yields real polynomials such that pn Ñ Re f and qn Ñ Im f inCpσpT qq as n Ñ 8, hence pn ` iqn Ñ f . We can thus extend the mapp ÞÑ ppT q to a linear isometry ΦT : f ÞÑ fpT q from CpσpT qq to BpXq. Bycontinuity, ΦT also satisfies (C4) and (C5) on CpσpT qq.

4.2. The spectral theorems 73

If there is another map Ψ : CpσpT qq Ñ BpXq satisfying (C1)-(C5), thenΨppq “ ppT q “ ΦT ppq for all polynomials by (C1), (C3) and (C4), so thatΨ “ ΦT by continuity and density.

We observe that we have actually shown uniqueness in then calss of linearand continuous maps Ψ : CpσpT qq Ñ BpXq fulfilling (C3) and (C4).

Lemma 4.12. Let T P BpZq for a Banach space Z and let p be a polyno-mial. Then σpppT qq “ ppσpT qq.

Proof. See Theorem 5.3 below or the exercises.

Corollary 4.13. Let T P BpXq be selfadjoint and f P CpσpT qq. Thenthe following asssertions hold.

(C6) If Tx “ λx for some x P X and λ P C, then fpT qx “ fpλqx.(C7) The operator fpT q is normal.(C8) σpfpT qq “ fpσpT qq (spectral mapping theorem).(C9) fpT q is selfadjoint if and only if f is real-valued.

Proof. Take a sequence of polynomials pn converging to f uniformly.Let Tx “ λx. Property (C6) holds for a polynomial since

ppT qx “nÿ

j“0

ajTjx “

nÿ

j“0

ajλjx “ ppλqx.

Using (C2), we obtain

fpT qx “ limnÑ8

pnpT qx “ limnÑ8

pnpλqx “ fpλqx.

From (C5) and (C4) we infer fpT qfpT q1 “ fpT qfpT q “ fpT qfpT q “fpT q1fpT q so that fpT q is normal.

We next show (C8). If µ R fpσpT qq, then g “ 1µ´f is an element of

CpσpT qq. Thus (C3) and (C4) yield

pµI ´ fpT qqgpT q “ gpT qpµI ´ fpT qq “ pgpµ1´ fqqpT q “ 1pT q “ I.

Hence, µ P ρpfpT qq. Let µ “ fpλq for some λ P σpT q. Then µn :“ pnpλqbelongs to σppnpT qq for all n P N by Lemma 4.12. As above, the operatorsµnI´pnpT q tend to µI´fpT q in BpXq. Suppose that µI´fpT q was invert-ible. Then also µnI´pnpT q would be invertible for large n by Theorem 1.27.This is impossible, and so µ P σpfpT qq.

For the last assertion, observe that fpT q “ fpT q1 if and only if pf´fqpT q “0 if and only if f ´ f “ 0, because ΦT is injective.

Corollary 4.14. Let n P N and T “ T 1 P BpXq with σpT q Ď R` (in thiscase one writes T “ T 1 ě 0). Then there is a unique selfadjoint operatorW P BpXq with σpW q Ď R` such that Wn “ T .

Proof. Consider wptq “ t1n for t P σpT q Ď R` and define W :“ wpT q.Then Wn “ wnpT q “ p1pT q “ T . Properties (C8) and (C9) imply that W “

W 1 ě 0. For the proof of the uniqueness of W , we refer to Korollar VII.1.16in [We] or the exercises.

We next use basic properties of orthogonal projections and orthonormalbases which are discussed in, e.g., Chapter 3 of [Sc2].

4.2. The spectral theorems 74

Theorem 4.15 (Compact case). Let X be a Hilbert space with dimX “

8, T P BpXq be compact and selfadjoint, and A be densely defined, closedand selfadjoint on X having a compact resolvent. Then the following asser-tions hold.

(a) i) There is an index set J P tH,N, t1, . . . , Nuˇ

ˇN P Nu and eigenvalues

λj ‰ 0, j P J , such that σpT q “ t0u 9Ytλjˇ

ˇ j P Ju Ď R where λj Ñ 0 asj Ñ8 if J “ N.

ii) There is an orthonormal basis of NpT qK “ RpT q consisting of eigen-vectors of T for the eigenvalues λj.

iii) The eigenspace EjpT q :“ NpλjI ´ T q is finite-dimensional and theorthogonal projection Pj onto EjpT q commutes with T , for each j P J .

iv) The sum T “ř

jPJ λjPj converges in BpXq.(b) i) We have σpAq“σppAq“tµn

ˇ

ˇn P Nu Ď R with |µn|Ñ8 as nÑ8.ii) There is an orthonormal basis of X consisting of eigenvectors of A.iii) The eigenspaces EnpAq “ NpµnI ´ Aq are finite dimensional and

the orthogonal projections Qn onto EnpAq satisfy Qn DpAq Ď DpAq andAQnx “ QnAx for all x P DpAq and n P N.

iv) The sum Ax “ř8n“1 µnQnx converges in X for all x P DpAq.

Proof. 1) Theorem 2.11, 2.16 and 4.7 show the assertions i) in (a) and(b), except for the claim that σpAq is infinite, and also that dimEjpT q anddimEnpAq are finite for all j and n. Let x, y P DpAq be eigenvectors of Afor eigenvalues µn ‰ µk. Then

µn px|yq “ pAx|yq “ px|Ayq “ µk px|yq,

so that px|yq “ 0. Similarly, one sees that EjpT q K EkpT q if j ‰ k. Bythe Gram-Schmidt procedure, each eigenspace EjpT q and EnpAq has anorthonormal basis of eigenvectors for λj ‰ 0 and µn, respectively. Theunion of these bases gives orthonormal sets BT and BA.

2) Let J “ N in a), the other cases are treated similarly. Observe thatBT Ď RpT q. Set 1j “ 1tλju P CpσpT qq and ϕn “ 11 ` . . . ` 1n for every

j, n P N. We then have 1jpT q2 “ 1

2j pT q “ 1jpT q. Moreover, (C4) and (C9)

imply that T1jpT q “ 1jpT qT , 1jpT q1 “ 1jpT q and

pλjI ´ T q1jpT q “ ppλj1´ p1q1jqpT q “ ppλj ´ λjq1jqpT q “ 0.

If λjv “ Tv for some v P X, we further deduce 1jpT qv “ 1jpλjqv “ v from(C6). As a result, 1jpT q is a selfadjoint projection onto EjpT q. For x P Xand y P Np1jpT qq, we then obtain p1jpT qx|yq “ px|1jpT qyq “ 0 so that1jpT q is orthogonal; i.e., 1jpT q “ Pj and ϕnpT q “ P1 ` . . . ` Pn for allj, n P N. We have shown iii) in (a).

Since TPj “ λjPj , the operator ϕnpT qT is a partial sum of the series inpart iv). Employing also (C2), we thus derive iv) from

‖T ´ ϕnpT qT‖ “ ‖pp1 ´ ϕnp1qpT q‖ “ ‖p1 ´ ϕnp1‖8 “ supjěn`1

|λj | ÝÑ 0,

as n Ñ 8. It also follows that ϕnpT qy P linBT converges to y as n Ñ 8

for all y P RpT q. Therefore, BT is an orthonormal basis of RpT q due to e.g.

Theorem 3.15 in [Sc2]. Finally, (2.1) shows that RpT q “ KNpT 1q “ NpT qK

because T “ T 1 and X is reflexive.

4.2. The spectral theorems 75

3) Fix t P ρpAqXR. Then Rpt, Aq1 “ Rpt, A1q “ Rpt, Aq by (4.2), and thisoperator is compact and has a trivial kernel. By 2), X “ NpRpt, AqqK pos-sesses an orthonormal basis of eigenvectors w of Rpt, Aq for the eigenvaluesλ ‰ 0. Proposition 1.20 yields the eigenvector v “ λRpt, Aqw of A for theeigenvalue µ “ t ´ λ´1. Since dimX “ 8 and the eigenspaces of Rpt, Aqare finite dimensional, A has infinitely many distinct eigenvalues; i.e., parti) in (b) is shown.

Let x P X be orthogonal to all eigenvectors of A. For the above v, weobtain 0 “ px|vq “ px|λRpt, Aqwq “ λ pRpt, Aqx|wq . Since the eigenvectorsw span X, we infer that Rpt, Aqx “ 0 and hence x “ 0. Consequently, BAis a basis of X and part ii) of (b) is true.

4) Let tvn,1, ¨ ¨ ¨ , vmnu be eigenvectors of A forming an orthonormal basisof EnpAq for n P N. From step 3) we then deduce

Qnx “mnÿ

j“1

px|vn,jq vn,j and x “8ÿ

n“1

Qnx

for all x P X. For x P DpAq it follows

QnAx “mnÿ

j“1

pAx|vn,jq vn,j “mnÿ

j“1

px|Avn,jq vn,j “mnÿ

j“1

px|µnvn,jq vn,j

“

mnÿ

j“1

px|vn,jq µnvn,j “mnÿ

j“1

px|vn,jq Avn,j “ AQnX.

We thus conclude

Anÿ

k“1

Qkx “nÿ

k“1

QkAx ÝÑ8ÿ

k“1

QkAx “8ÿ

k“1

AQkx “8ÿ

k“1

µkQkx,

as nÑ8, so that the closedness of A yields the last assertion.

Remark 4.16. a) In the above proof we also obtain that

Ax “8ÿ

n“1

µn

mnÿ

j“1

px|vn,jq vn,j

for all x P DpAq. An analogous result holds for T .

b) Let T be selfadjoint and compact such that NpT q is separable. Lettzk

ˇ

ˇ k P J0u be an orthonormal basis for NpT q, where J0 Ď Z´ could beempty. Denote by λl the non-zero eigenvalues of T (repeated accordingto their multiplicity) with corresponding orthonormal basis of eigenvectorstwl

ˇ

ˇ l P J1u. The union tbjˇ

ˇ j P J 1u of tzkˇ

ˇ k P J0u and twlˇ

ˇ l P J1u is anorthonormal basis of X. By e.g. Theorem 3.18 in [Sc2], the map

Φ : X Ñ `2pJ 1q; Φx “`

px|bjq˘

jPJ 1,

is unitary with Φ´1ppξjqjPJ 1q “ř

jPJ 1 ξjbj . Moreover, the transformed oper-

ator ΦTΦ´1 acts on `2pJ 1q as the multiplication operator

ΦTΦ´1pξjq “ ΦTÿ

jPJ 1

ξjbj “ Φÿ

jPJ 1

λjξjbj “ pλjξjqjPJ 1 ,

4.2. The spectral theorems 76

where λj :“ 0 if j P J0. Hence, ΦTΦ´1 is represented as infinite diago-nal matrix with diagonal elements λj . Analogous results hold for A fromTheorem 4.15, see e.g. Theorems 4.5.1-4.5.3 in [Tr]. ♦

If one wants to extend part a) of the above remark to the non-compactcase one is led to the so-called spectral measures, see the exercises. Wegeneralize below part b).

Theorem 4.17 (Multiplicator representation, bounded case). Let T P

BpXq be selfadjoint on a separable Hilbert space X. Then there is a σ–finitemeasure space pΩ,A, µq, a measurable function h : Ω Ñ σpT q and a unitaryoperator U : X Ñ L2pµq such that

Tx “ U´1hUx for all x P X.

Proof (partly sketched). 1) Let v1 P Xzt0u. We define the linearsubspaces

Y1 :“ tfpT qv1

ˇ

ˇ f P CpσpT qqu and X1 “ Y 1

of X. Since TfpT qv1 “ pp1fqpT qv1 P Y1 for every f P CpσpT qq, we obtainTY1 Ď Y1 and so TX1 Ď X1. We introduce the map

ϕ1 : CpσpT qq Ñ C; ϕ1pfq “ pfpT qv1|v1q ,

which is linear and bounded because |ϕ1pfq| ď ‖fpT q‖‖v1‖2 “ ‖f‖8‖v1‖2

due to (C2). If f ě 0, then σpfpT qq Ď R` by (C8). So we can deduce fromCorollary 4.14 that

pfpT qv1|v1q “`

fpT q12v1

ˇ

ˇfpT q12v1

˘

“ ‖fpT q12v1‖2 ě 0.

The Riesz representation theorem of CpσpT qq˚ now gives a positive measureµ1 on BpσpT qq such that

ϕ1pfq “

ż

σpT qf dµ1,

for all f P CpσpT qq Ď L2pµ1q, see Theorem 2.14 of [Ru1]. For x “ fpT qv1 P

Y1, we define V1x :“ f P L2pµ1q. We compute

‖V1x‖2L2pµ1q

“

ż

σpT q|f |2 dµ1 “ ϕ1pffq “

`

pffqpT qv1|v1q

“`

fpT q1fpT qv1|v1q “ pfpT qv1|fpT qv1q “ ‖x‖2X .

In particular, if x “ fpT qv1 “ gpT qv1 for some g P CpσpT qq, then

‖f ´ g‖22 “ ‖pfpT q ´ gpT qqv1‖2

X “ 0,

and so f “ g in L2pµ1q. As a result, V1 : Y1 Ñ L2pµ1q is a linear isometricmap and can be extended to a linear isometry U1 : X1 Ñ L2pµ1q.

Observe that CpσpT qq Ď RpU1q. Riesz’ theorem also yields that µ1 isregular.1 Theorem 3.14 in [Ru1] then shows that CpσpT qq is dense in L2pµ1q,

1A positive measure µ is called regular, if if satisfies

µpBq “ inftµpOqˇ

ˇB Ď O, O is openu “ suptµpKqˇ

ˇK Ď B,K is compactu

for all B in the corresponding Borel σ–algebra.

4.2. The spectral theorems 77

so that U1 has dense range. Hence, the isometry U1 is bijective and thusunitary by e.g. Proposition 5.52 in [Sc2]. Finally, we compute

U1TfpT qv1 “ V1pp1fqpT qv1 “ p1f “ p1U1fpT qv1,

for f P CpσpT qq. By density, it follows Tx “ U´11 p1U1x for all x P X1.

2)2 We are done if there is an v1 P X such that X1 “ X. In general thisis not true. Using Zorn’s Lemma (see Theorem I.2.7 of [DS1]), we insteadfind an orthonormal system of spaces Xn as in step 1) which span X. Tothat aim, we introduce the collection E of all sets E having as elements atmost countably many closed subspaces Xj Ă X of the type constructed instep 1) such that Xi K Xj for all Xi ‰ Xj in E. The system E is orderedvia inclusion of sets. Let C be a chain in E ; i.e., a subset of E such thatE Ď F or F Ď E for all E,F P C. We put C “

Ť

EPC E. Cleary, E Ď Cfor all E P C. Let Y,Z P C. Then Y and Z are closed subspaces of X asconstructed in 1) and there are sets E,F P C such that Y P E and Z P F .We may assume that E Ď F and so X,Y P F . The subspaces Y and Zare thus orthogonal (if Y ‰ Z). As a result, C contains pairwise orthogonalsubspaces of X. If x K y have norm 1, then x ´ y2 “ x2 ` y2 “ 2.The separability of X then implies that at most countably many subspacesbelong to C, so that C is an element of E and hence an upper bound of C.Zorn’s Lemma now gives a maximal element M “ tXj

ˇ

ˇ j P Ju in E , whereJ Ă N and Xj are pairwise orthogonal subspaces as constructed in step 1).

Assume that there is a vector v P X being orthogonal to all Xj . Let Zbe the closed span of all vectors fpT qv with f P CpσpT qq. Let g P CpσpT qqand x “ gpT qvi P Xi for some i P J , where vi generates Xi as in step 1). Wethen obtain

px|fpT qvq “ pfpT q1gpT qvi|vq “ ppfgqpT qvi|vq “ 0

since pfgqpT qvi P Xi. By density, it follows that Z is orthogonal to all Xj

and thus MYtZu P E . The maximality on M now implies that Z belongs toM , so that v “ 0. Consequently, X is the closed linear span of the elementsin the orthogonal subspaces Xj . (One writes X “

À

jPJ Xj .) We now define

Ω “ď

jPJ

σpT q ˆ tju Ď R2, A “ BpΩq, µpAq “ÿ

jPJ

µjpAjq

with Aj ˆ tju “ AX pσpT q ˆ tjuq and

hpλˆ tjuq “ λ, Ux “ÿ

jPJ

Ujxj ,

where λˆ tju P Ω, Uj is given as in step 1), and x “ř

jPJ xj and xj P Xj .It can be seen that Ω, A, µ, h and U satisfy the assertions.

We add an observation to the above proof. Let λ P σpT qzσppT q. ThenΛ “

Ť

jPJtλˆ tjuu Ă Ω is a µ–null set. In fact, otherwise the characteristic

function f of any subset of Λ with measure in p0,8q would be a non-zeroelement of L2pµq. Hence, x “ U´1f ‰ 0 would be an eigenvector of T forthe eigenvalue λ, since Tx “ U´1λf “ λx. We further note that in the proof

2Not shown in the lectures.

4.2. The spectral theorems 78

of Theorem 4.17 one could also take Ω “Ť

jPJ σpTjqˆtju, where Tj “ T |Xj .

It can be shown that σpT q “Ť

jPJ σpTjq.The above representation of bounded selfadjoint operators as multiplica-

tion operators now leads to a multiplication representation and to a Bb–functional calculus for (possibly) unbounded selfadjoint operators A. HereBbpσpAqq is the Banach space of bounded Borel functions on σpAq endowedwith the supremum norm. We use this space instead of L8pσpAqq to avoidcertain technical problems. Set rλpzq “ pλ´ zq

´1 for z P Cztλu.Theorem 4.18 (Unbounded A). Let A be a closed, densely defined and

selfadjoint operator on a separable Hilbert space X. Then the followingassertions hold.

(a) There is a σ–finite measure space pΩ,A, µq, a measurable functionh : Ω Ñ σpAq and a unitary operator U : X Ñ L2pµq such that

DpAq “ tx P Xˇ

ˇhUx P L2pµqu and Ax “ U´1hUx.

(b) There is a contractive map ΨA : BbpσpAqq Ñ BpXq; ΨT pfq “ fpAq,satisfying (C1) and (C3)–(C5), where p1pT q “ T in (C3) is replaced byrλpAq “ Rpλ,Aq for λ P ρpAq. Moreover, if fn P BbpσpAqq are uniformlybounded and converge to f P BbpσpT qq pointwise, then fnpAqx Ñ fpAqx asn Ñ 8 for all x P X. Finally, for x P DpAq and f P BbpσpAqq the vectorfpAqx belongs to DpAq and AfpAqx “ fpAqAx.

Proof. a) We additionally assume that σpAq ‰ R and fix t P ρpAqXR.3

Then Rpt, Aq P BpXq is selfadjoint and can be represented as Rpt, Aq “U´1mU on a space L2pΩ, µq as in Theorem 4.17. Recall that Proposi-tion 1.20 yields σpAq “ t´ rσpRpt, Aqqzt0us´1. Set

hpλˆ tjuq “ t´1

mpλˆ tjuq“ t´

1

λP σpAq,

for j P J and λ P σpRpt, Aqqzt0u. The sets t0 ˆ tjuu have µ–measure 0 inview of the obervation before the theorem, due to the injectivity of Rpt, Aq.We can thus extend h by 0 to a measurable function on Ω.

Let x P DpAq. We put y “ tx´Ax P X. Since x “ Rpt, Aqy “ U´1mUy,we obtain

hUx “ hmUy “ ptm´ 1qUy P L2pµq,

U´1hUx “ tU´1mUy ´ y “ tx´ y “ Ax.

If x P X satisfies hUx P L2pµq, then we put y “ U´1pt1 ´ hqUx P X andobtain mUy “ ptm ´mhqUx “ Ux. Therefore, x “ U´1mUy “ Rpt, Aqybelongs to DpAq, and part (a) is proved.

b) We define ΨA : f ÞÑ fpAq by

fpAqx “ U´1pf ˝ hqUx

for f P BbpσpAqq and x P X. We further set Mfϕ “ pf ˝ hqϕ for ϕ P L2pµq.It is straightforward to check that fpAq P BpXq, ΨT is linear, 1pAq “ I and(C5) is true. Let λ P ρpAq. We have

hUrλpAqx “ hprλ ˝ hqUx “ hpλ1´ hq´1Ux P L2pµq

3If σpAq “ R, we instead take t “ i and use below the version of Theorem 4.17 forthe normal operator Rpi, Aq given in e.g. Satz VII.1.25 in [We].

4.2. The spectral theorems 79

for all x P X. So part (a) yields that rλpAqX Ď DpAq and pλI´AqrλpAq “ I.Similarly, one sees that rλpAqpλx ´ Axq “ x for all x P DpAq, and so (C3)is shown. The contractivity follows from ‖fpAq‖ “ ‖Mf‖ ď ‖f‖8. Forproperty (C4) we observe that

pfgqpAqx “ U´1pf ˝hqpg ˝hqUx “ U´1pf ˝hqUU´1pg ˝hqUx “ fpAqgpAqx,

where f, g P BbpσpAqq and x P X.Let f, fn P BbpσpAqq be uniformly bounded by c such that fn Ñ f point-

wise as nÑ8. For every x P X, we have fnpAqx´ fpAqx “ U´1ppfn´ fq ˝hqUx. Since pfn ´ fq ˝ h Ñ 0 pointwise and |ppfn ´ fq ˝ hqUx| ď 2c |Ux|,Lebesgue’s convergence theorem shows that ppfn ´ fq ˝ hqUx tends to 0 inL2pµq and so fnpAqxÑ fpAqx in X as nÑ8.

Let x P DpAq. The above results yield that

g :“ hpf ˝ hqUx “ pf ˝ hqUU´1hUx “ pf ˝ hqUAx P L2pµq.

On the other hand, g “ hUU´1pf ˝hqUx “ hUfpAqx. Part (a) thus impliesthat fpAqx P DpAq and

AfpAqx “ U´1hUfpAqx “ U´1g “ U´1pf ˝ hqUAx “ fpAqAx.

We conclude with one of the most important applications of the abovetheorem.

Example 4.19 (Schrodinger’s equation). Let X be a Hilbert space andH be a closed, densely defined and selfadjoint operator on X. For a givenu0 P DpHq we claim that there is exactly one function u P C1pR, Xq XCpR, rDpHqsq solving

ddtuptq “ ´iHuptq, t P R, up0q “ u0. (4.5)

(H is called Hamiltonian.) The solution is given by uptq “ T ptqu0 for unitaryoperators T ptq on X satisfying T p0q “ I, T ptqT psq “ T psqT ptq and T ptq´1 “

T p´tq for t, s P R. Moreover, t ÞÑ T ptqx P X continuous for t P R and allx P X. An example for this setting is X “ L2pR3q and H “ ´p∆ ` b

|x|2 q

with DpHq “W 2,2pR3q, see Example 4.10.Proof. For t P R, we consider the bounded function ftpξq “ e´itξ, ξ P R.

Theorem 4.18 allows us to define T ptq “ ftpHq P BpXq. Since f0 “ 1 andftfs “ ft`s, we obtain T p0q “ I and T ptqT psq “ T psqT ptq for t, s P R. Withs “ ´t it follows that T ptq has the inverse T p´tq. Moreover, T ptq is unitarysince T ptq1 “ ftpHq “ f´tpHq “ T p´tq. Because of ‖ft‖8 “ 1 and thecontinuity of t ÞÑ ftpξq for fixed ξ P R, the function R Q t ÞÑ T ptqz P X iscontinuous for each x P X.

Let u0 P DpHq. We set y “ τu0 ´ Hu0 for some τ P ρpHq, so thatu0 “ Rpτ,Hqy “ rτ pHqy. We then obtain

1

t´ spT ptqu0 ´ T psqu0q “

1

t´ spftpHq ´ fspHqqrτ pHqy

“

ˆ

1

t´ spft ´ fsqrτ

˙

pHqy “: gt,spHqy

for all t ‰ s. Observe that gt,spξq Ñ´iξτ´ξfspξq “: mpξqfspξq as t Ñ s for

all ξ P σpHq and ‖gt,s‖8 ď ‖m‖8 “ supξPσpHq|ξ

τ´ξ | ă 8 for all t ‰ s. So

4.2. The spectral theorems 80

Theorem 4.18 shows that there existsddtT ptqu0 “ mpHqT ptqpτu0 ´Hu0q “ T ptqmpHqy.

We further compute

mpT qy “ Upm ˝ hqU´1y “ Upp´ip1rτ q ˝ hqU´1y

“ ´U ihU´1Uprτ ˝ hqU´1y “ ´iHRpτ,Hqy “ ´iHu0

by means of Theorem 4.18. Hence, we arrive atddtT ptqu0 “ ´iT ptqHu0 “ ´iHT ptqu0,

using Theorem 4.18 once more. Due to these equations, u “ T p¨qu0 belongsto C1pR`, Xq X CpR`, rDpHqsq and solves (4.5).

Let v P C1pR`, Xq X CpR`, rDpHqsq be another solution of (4.5). For0 ď s ď t and h ‰ 0 we compute1hpT pt´ s´ hqvps` hq ´ T pt´ sqvpsqq ´ T pt´ sqpv

1psq ` iHvpsqq

“ T pt´ s´ hq`

1hpvps` hq ´ vpsqq ´ v

1psq˘

` pT pt´ s´ hq ´ T pt´sqqv1psq

´ 1´hpT pt´ s´ hq ´ T pt´ sqqvpsq ´ T pt´ sqiHvpsq ÝÑ 0

as h Ñ 0. For any y P X we thus obtain dds pT pt´ sqvpsq|yq “ 0, since

v1 “ ´iHv. Consequently,

pT ptqx|yq “ pT ptqvp0q|yq “ pT p0qvptq|yq “ pvptq|yq ,

which gives uptq “ vptq for all t ě 0. Thus the ‘strongly continuous unitarygroup’ pT ptqqtPR solves (4.5) uniquely. l

CHAPTER 5

Holomorphic functional calculi

We come back to the case of Banach spaces X and Y . We want tointroduce functional calculi for non-selfadjoint operators on X, using nowcomplex curve integrals.

5.1. The bounded case

Let U Ď C be open, g : U Ñ Y be holomorphic (i.e., complex dif-ferentiable) and Γ Ď U be a piecewise C1 curve with parametrizationγ : ra, bs Ñ U . This means that there are a “ a0 ă a1 ă ¨ ¨ ¨ ă am “ b suchthat γ P C1prak´1, aks,Cq for all k P t1, . . . ,mu, where γ does not need tobe continuous. We write Γ “ Γ1 Y ¨ ¨ ¨ Y Γn with Γk “ γkprak´1, aksq anddefine the curve integral

ż

Γg dz “

ż b

agpγptqqγ1ptqdt :“

mÿ

k“1

ż ak

ak´1

gpγptqqγ1ptqdt

as a Banach space-valued Riemann integral (having the same definition,results and proofs as for Y “ R in Analysis 1). Using Riemann sums, oneeasily checks that T

ş

Γ g dz “ş

Γ Tg dz for all T P BpY,Zq and Banach spacesZ. The index of a closed curve (i.e., γpaq “ γpbq) is given by

npΓ, zq “1

2πi

ż

Γ

dw

w ´ z

for all z P CzΓ. The index is the number of times that Γ winds around z,counted with orientation.

If Γ is closed and npΓ, zq “ 0 for all z R U , then Cauchy’s integral theoremand formula

ż

Γg dz “ 0, (5.1)

npΓ, zqgpzq “1

2πi

ż

Γ

gpwq

w ´ zdw, (5.2)

are valid for all z P UzΓ. This fact is shown for Y “ C in Theorems IV.5.4and IV.5.7 of [Co1]. For a general Banach space Y , the formulas (5.1) and(5.2) are thus satisfied by functions z ÞÑ xgpzq, y˚y for every y˚ P Y ˚. Hence,xş

Γ g dz, y˚y “ 0 for all y˚ P Y ˚ so that a corollary of the Hahn–Banachtheorem yields (5.1) in Y . Similarly one deduces (5.2).

For compact, non-empty subsets K Ď C, we introduce the space

HpKq “tf : Dpfq Ñ Cˇ

ˇK Ď Dpfq Ď C, Dpfq is open, f is holomorphicu.

Let K Ď U Ď C, K be compact, and U be open. By Proposition VIII.1.1 in[Co1] and its proof there exists an admissible curve Γ for K and U (or, in

81

5.1. The bounded case 82

UzK) which means that Γ Ď UzK is piecewise C1, npΓ, zq “ 1 for all z P Kand npΓ, zq “ 0 for all z P CzU .

Let T P BpXq, f P HpσpT qq, and Γ be admissible for σpT q and Dpfq. Wethen define

fpT q :“1

2πi

ż

ΓfpλqRpλ, T qdλ P BpXq. (5.3)

This integral exists in the Banach space BpXq since λ ÞÑ fpλqRpλ, T q isholomorphic on ρpT q X Dpfq Ě Γ. Writing Rpλ, T q as “ 1

λ´T ” one sees the

similarity of (5.3) and (5.2), but hereRpλ, T q does not exist on σpT q, whereasin (5.2) the function w ÞÑ 1

w´z is defined on Cztzu.If Γ1 is another admissible curve for σpT q and Dpfq, then we set Γ2 “

Γ Y p´Γ1q, where “´” denotes the inversion of the orientation. We thenhave

npΓ2, zq “ npΓ, zq ´ npΓ1, zq “

#

1´ 1 “ 0, z P σpT q,

0´ 0 “ 0, z P CzDpfq.

So we can apply (5.1) on U “ DpfqzσpT q obtaining

0 “

ż

Γ2fpλqRpλ, T qdλ “

ż

ΓfpλqRpλ, T q dλ´

ż

Γ1fpλqRpλ, T qdλ.

Consequently, (5.3) does not depend on the choice of the admissible curve.We recall that rλpzq “

1λ´z and p1pzq “ z for λ, z P C with λ ‰ z.

Theorem 5.1. Let T P BpXq for a Banach space X. Then the map

ΦT : HpσpT qq Ñ BpXq, f ÞÑ fpT q,

defined by (5.3) is linear and satisfies

(H1) ‖fpT q‖ ď c supλPΓ|fpλq| for a constant c “ cpΓ, T q ą 0,(H2) 1pT q “ I, p1pT q “ T , rλpT q “ Rpλ, T q,(H3) fpT qgpT q “ gpT qfpT q “ pfgqpT q,(H4) fpT q˚ “ fpT ˚q,(H5) if fn Ñ f uniformly on compact subsets of Dpfq, then fnpT q Ñ fpT q

in BpXq as nÑ8,

for all λ P ρpT q and f, g, fn P HpσpT qq with Dpfnq “ Dpfq for every n P N.ΦT is the only linear map from HpσpT qq to BpXq satisfying (H1)–(H3).

For a polynomial p, the operators ppT q in (5.3) and in (4.4) coincide. If Xis a Hilbert space and T “ T 1, the above ΦT is the restriction of the map ΦT

from Theorem 4.11.

Proof. It is clear that f ÞÑ fpT q is linear. Property (H1) follows from

‖fpT q‖ ď 1

2π`pΓq sup

λPΓ‖Rpλ, T q‖ sup

λPΓ|fpλq| “: cpΓ, T q sup

λPΓ|fpλq|.

Replacing here fpT q by fpT q ´ fnpT q “ pf ´ fnqpT q we also deduce (H5).To check (H4), we recall that σpT q “ σpT ˚q and Rpλ, T q˚ “ Rpλ, T ˚q fromTheorem 1.24. Hence,

fpT q˚ “1

2πi

ż

ΓfpλqRpλ, T q˚ dλ “ fpT ˚q.

We next show (H3). We choose a bounded open set U Ď C such thatσpT q Ď U Ď U Ď Dpfq X Dpgq and admissible curves Γf in UzσpT q and

5.1. The bounded case 83

Γg in pDpfq X DpgqqzU . We then have npΓf , µq “ 0 for all µ P Γg Ď CzUand npΓg, λq “ 1 for all λ P Γf Ď U . Using the resolvent equation, Fubini’stheorem in BpXq (see e.g. Theorem X.6.16 in [AE]) and (5.2) in C, wecompute

fpT qgpT q “1

2πi

ż

Γf

fpλqRpλ, T q1

2πi

ż

Γg

gpµqRpµ, T qdµdλ

“

ˆ

1

2πi

˙2 ż

Γf

ż

Γg

fpλqgpµq1

µ´ λpRpλ, T q ´Rpµ, T qqdµdλ

“1

2πi

ż

Γf

fpλqRpλ, T q1

2πi

ż

Γg

gpµq

µ´ λdµdλ

`1

2πi

ż

Γg

gpµqRpµ, T q1

2πi

ż

Γf

fpλq

λ´ µdλdµ

“1

2πi

ż

Γf

fpλqgpλqRpλ, T q dλ “ pfgqpT q.

This identity also yields pfgqpT q “ pgfqpT q “ gpT qfpT q.To check (H2), we take f “ 1 with Dpfq “ C. We choose Γ0 “

BBp0, 2‖T‖q. Theorem 1.16 then leads to

1pT q “1

2πi

ż

Γ0

Rpλ, T qdλ “1

2πi

ż

Γ0

8ÿ

n“0

Tnλ´n´1 dλ

“

8ÿ

n“0

Tn1

2πi

ż

Γ0

λ´n´1 dλ “ I,

since the series converges in BpXq uniformly on Γ0 andş

Γ0λ´m dλ is equal to

2πi if m “ 1 and equal to 0 for m P Zzt1u. The property p1pT q “ T is shownsimilarly. For λ P ρpT q, consider fλpzq “ λ´ z. The previous results implyfλpT q “ λI ´ T and fλpT qrλpT q “ rλpT qfλpT q “ prλfλqpT q “ 1pT q “ I sothat rλpT q “ Rpλ, T q.

Let Ψ : HpσpT qq Ñ BpXq be any linear map satisfying the assertions(H1)–(H3). The linearity, (H2) and (H3) imply that Ψppq “ ppT q “ ΦT ppqfor every polynomial p. If f P HpσpT qq and Γ Ď DpfqzσpT q is admissible,then there are polynomials pn converging uniformly to f on the compact setΓ. Hence, pnpT q Ñ Ψpfq by (H1) and thus Ψ “ ΦT . The final assertion isshown similarly.

Example 5.2. Let E “ CpKq for a compact set K Ď Rd and let m P

CpKq. We define Mϕ “ mϕ for ϕ P E. Proposition 1.14 shows thatM P BpEq, σpMq “ mpKq, and Rpλ,Mqϕ “ 1

λ´mϕ for all λ P ρpMq.

Let f P HpmpKqq, Γ be an admissible curve in DpfqzmpKq, ϕ P E, andx P K. Using that the map ψ ÞÑ ψpxq is continuous and linear from E to Cand Cauchy’s formula (5.2) for µ “ mpxq P mpKq, we compute

pfpMqϕqpxq “1

2πi

ż

ΓfpλqRpλ,Mqϕdλpxq “

1

2πi

ż

ΓfpλqpRpλ,Mqϕqpxq dλ

“1

2πi

ż

Γ

fpλq

λ´mpxqdλϕpxq “ fpmpxqqϕpxq.

5.1. The bounded case 84

As a result, fpMqϕ “ pf ˝mqϕ is also a multiplication operator. ♦

Theorem 5.3. Let T P BpXq and f P HpσpT qq. We then have the spectralmapping theorem

σpfpT qq “ fpσpT qq.

Proof. Let µ R fpσpT qq. After possibly shrinking Dpfq, the map g “1

µ1´f belongs to HpσpT qq. The properties of the calculus yield

gpT qpµI ´ fpT qq “ pµI ´ fpT qqgpT q “ pgpµ1´ fqqpT q “ I,

so that µ P ρpfpT qq.

Conversely, let µ “ fpλq for some λ P σpT q. We set gpzq “ fpzq´µz´λ for

z P Dpfqztλu and gpλq “ f 1pλq. Since g is bounded, it is holomorphic onDpfq by Riemann’s theorem on removable singularities. Moreover, gpzqpz´λq “ fpzq ´ µ for all z P Dpfq and so

pλI´T qgpT q “ gpT qpλI´T q “ pgpλ1´p1qqpT q “ pµ1´fqpT q “ µI´fpT q.

Because the operator pλI ´ T q is not surjective or not injective, µI ´ fpT qhas the respective properties. Hence, µ is contained in σpfpT qq.

The next result can in particular be used to solve linear ordinary differ-ential equations governed by matrices A.

Example 5.4. Let A P BpXq. For t P R we set ftpzq “ etz, z P C, anddefine etA “ ftpAq P BpXq. As in Example 4.19, one sees that

ept`sqA “ etAesA “ esAetA, e0A “ I, retAs´1 “ e´tA

for all t, s P R. Moreover, t ÞÑ etA belongs to C1pR,BpXqq with

ddte

tA “ AetA “ etAA.

Hence, the map uptq “ etAu0 is the unique solution in C1pR, Xq of

ddtuptq “ Auptq, t P R, up0q “ u0,

where u0 P X is given. Theorem 5.3 further yields

r`

etA˘

“ max

|µ|ˇ

ˇµ P σpetAq “ etσpAq(

“ max

etReλˇ

ˇλ P σpAq(

“ etspAq

for the spectral bound spAq :“ maxtReλˇ

ˇλ P σpAqu. Therefore, if spAq ă 0

(i.e., σpAq Ď tλˇ

ˇ Reλ ă 0u), then we deduce from Theorem 1.16 that

1 ą rpeAq “ limnÑ8

‖peAqn‖1n “ limnÑ8

‖enA‖1n.

So we can fix an index N P N with ‖eNA‖ “: q ă 1. Writing any given t ě 0as t “ kN ` τ for some k P N0 and 0 ă τ ď N we estimate

‖etA‖ “ ‖`

eNA˘k

eτA‖ ď qk ‖eτA‖ ď max0ďτďN

‖eτA‖ exp´

Nkln q

N

¯

ďMe´wt

where w :“ ´ ln qN ą 0 and M :“ max0ďτďN‖eτA‖ e|ln q|. So spectral infor-

mation on the given operator A implies the exponential decay

‖uptq‖ ďMe´wt‖x‖, t ě 0,

of the solutions u. ♦

5.1. The bounded case 85

Let S P BpXq and P “ P 2 P BpXq be a projection with SP “ PS. SetX1 “ RpP q and X2 “ NpP q. Then, X “ X1 ‘ X2 by e.g. Lemma 2.16 of[Sc2]. Moreover, if y “ Px P RpP q, then Sy “ SPx “ PSx P RpP q. Ifx P NpP q, then PSx “ SPx “ 0 and so Sx P NpP q. As a result, S leavesinvariant X1 and X2 and the restrictions S|Xj P BpXjq are well defined.

Theorem 5.5 (Spectral projection). Let T P BpXq and σpT q “ σ1 9Yσ2

for two disjoint closed sets σj ‰ H in C. Then there is a projection P P

BpXq with fpT qP “ PfpT q for all f P HpσpT qq such that σpTjq “ σj forj P t1, 2u, where Tj “ T|Xj P BpXjq, X1 “ RpP q and X2 “ NpP q. Moreover,

X “ X1 ‘X2 and Rpλ, Tjq “ Rpλ, T q|Xj for λ P ρpT q “ ρpT1q X ρpT2q. Wefurther have

P “1

2πi

ż

Γ1

Rpλ, T qdλ, (5.4)

where Γ1 is an admissible curve for σ1 and any open set U1 Ě σ1 such thatU1 X σ2 “ H.

Proof. There are open sets Uj with U1 X U2 “ H and σj Ď Uj forj P t1, 2u. Define h P HpσpT qq by h “ 1 on U1 and h “ 0 on U2. Weset P “ hpT q P BpXq. We then deduce P 2 “ h2pT q “ hpT q “ P andfpT qP “ PfpT q for all f P HpσpT qq from (H3) and (H2). The aboveremarks show that X “ X1 ‘X2 and that the operators Tj “ T|Xj P BpXjq

are well defined.The formula (5.4) follows by choosing Γ “ Γ1YΓ2, where Γj are admissible

curves for σj and Uj for j P t1, 2u. Let λ R σ1. We may shrink U1 so thatλ R U1 since P does not depend on the choice of Γ and thus not on thechoice of U1. We define gpzq “ 1

λ´z for z P U1 and gpzq “ 0 for z P U2. Then

g P HpσpT qq and

gpT qpλI ´ T q “ pλI ´ T qgpT q “ ppλ1´ p1qgqpT q “ hpT q “ P.

Setting R “ gpT q|X1P BpX1q, we thus obtain

RpλIX1 ´ T1q “ pλIX1 ´ T1qR “ IX1 .

This means that λ P ρpT1q, and so σpT1q Ď σ1. Similarly, one sees thatσpT2q Ď σ2. In particular, σpT1q and σpT2q are disjoint. Let λ P ρpT1q X

ρpT2q. Given x P X, we have unique x1 P X1 and x2 P X2 such thatx “ x1`x2. If λx´Tx “ 0, then 0 “ λx1´T1x1`λx2´T2x2 P X1‘X2 sothat xj belongs to NpλI ´ Tjq “ t0u for j P t1, 2u; i.e., x “ 0. Given y P X,we define xj “ Rpλ, Tjqyj P Xj for j P t1, 2u. Setting x “ x1`x2, we derive

λx´ Tx “ λx1 ´ T1x1 ` λx2 ´ T2x2 “ y1 ` y2 “ y.

We have proved that λ P ρpT q, Rpλ, T qy “ Rpλ, T1qy1 ` Rpλ, T2qy2,Rpλ, T q|Xj “ Rpλ, Tjq, and

σpT q “ σ1 9Yσ2 Ď σpT1q 9YσpT2q Ď σ1 9Yσ2,

which implies that σpTjq “ σj for j P t1, 2u.

We use the above concept to refine the results in Example 5.4 about thelong-term behavior of etA.

5.2. Sectorial operators 86

Example 5.6 (Exponential dichotomy). In the setting of Example 5.4,assume that σpAq X iR “ H. Hence, σpAq “ σ1 9Yσ2 where σ1 Ď tλ PCˇ

ˇ Reλ ă 0u and σ2 Ď tλ P Cˇ

ˇ Reλ ą 0u. Let P be the spectral projectionof A for σ1 and define A1 and A2 as the restrictions of A to X1 “ RpP q andX2 “ NpP q, respectively, as in Theorem 5.5. Then petAq|Xj “ etAj : Xj Ñ

Xj and there are constants δ,N ą 0 such that

etA1 ď Ne´δt and e´tA2 ď Ne´δt, t ě 0.

In other words, X can be decomposed into etA-invariant subspaces on whichetA decays exponentially in forward and in backward time, respectively.Proof. Let Γ1 be given as in Theorem 5.5. For x P X1 we compute

etAx “ etAPx “ pfthqpAqx “1

2πi

ż

Γ1

etλRpλ,Aqx dλ

“1

2πi

ż

Γ1

etλRpλ,A1qx dλ “ etA1x,

where ftpλq “ etλ for t P R and h is given by the proof of Theorem 5.5.In the same way one derives etAx “ etA2x for all x P X2 and t ě 0. SinceσpA1q “ σ1, we have spA1q ă 0 and so Example 5.4 shows that ‖etAx1‖ ďMe´ωt‖x1‖ for all t ě 0 and x1 P X1 and some constants M,ω ą 0.

Moreover, σpA2q “ σ2 and so sp´A2q ă 0. Note that the curve Γ “

t´λˇ

ˇλ P Γu is admissible for σp´Aq “ ´σpAq. Substituting µ “ ´λ, weconclude that

e´tA “1

2πi

ż

Γe´tλpλI ´Aq´1 dλ “

1

2πi

ż

ΓetµpµI ´ p´Aqq´1 dµ “ etp´Aq

for all t P R. For x2 P X2 we thus obtain

e´tA2x2 “ e´tAx2 “ etp´Aqx2 “ etp´A2qx2

so that ‖e´tA2x2‖ ďM 1e´ω1t‖x2‖ for all t ě 0 and some M 1, ω1 ą 0. l

5.2. Sectorial operators

We extend the above results to certain unbounded operators A, restrictingourselves to the exponential etA.1 For φ P p0, πs we define the open sector

Σφ “ tλ P Czt0uˇ

ˇ |arg λ| ă φu.

We also set Σπ2 “: C` and C´ “ ´C`. Note that Σπ “ CzR´.

Definition 5.7. A closed operator A is called sectorial of angle φ P p0, πsif there is a constant K ą 0 such that Σφ Ď ρpAq and

‖Rpλ,Aq‖ ď K

|λ|, λ P Σφ.

In the literature several small variations of the above definition are used.Note that a sectorial operator of angle φ is also sectorial of angle φ1 P p0, φq.In applications often arise operators A such that A´ωI is sectorial for someω P R, cf. Remark 5.13.

1See [KW] for a detailed study of a corresponding functional calculus.

5.2. Sectorial operators 87

Example 5.8. Let X be a Hilbert space and A be closed, densely definedand selfadjoint on X. We further suppose that σpAq Ď R´. Then A issectorial of any angle φ P pπ2 , πq.

Proof. Let φ P pπ2 , πq and λ P Σφ. Since Rpλ,Aq1 “ Rpλ,Aq, theoperator Rpλ,Aq is normal. Propositions 4.3 and 1.20 then yield

‖Rpλ,Aq‖ “ rpRpλ,Aqq “ dpλ, σpAqq´1 ď dpλ,R´q´1 “

#

1|λ| , Reλ ě 0,

1|Imλ| , Reλ ă 0.

If Reλ ă 0, we can write λ “ |λ|e˘iθ for some θ P pπ2 , φq. We then have|Imλ||λ| “ | sin θ| ě sinφ ą 0, and thus

‖Rpλ,Aq‖ ď1

sinφ

|λ|“:

Kφ

|λ|, λ P Σφ. l

Note that Kφ Ñ8 as φÑ π in the above example.

Example 5.9. Let X “ Cpr0, 1sq and Au “ u2 with DpAq “ tu P

C2pr0, 1sqˇ

ˇup0q “ up1q “ 0u. Then A is sectorial of any angle φ ă π.Proof. We first recall from Example 3.40 that σpAq “ σppAq “

t´π2k2ˇ

ˇ k P Nu. Let λ P Σπ and λ “ µ2 for some µ P C with Reµ ą 0. Letf P X. There exists a unique function u P DpAq and λu´Au “ f ; i.e.,

u P C2pr0, 1sq, u2 “ µ2u´ f on r0, 1s, up0q “ up1q “ 0.

The solution of this ordinary boundary value problem is given by

uptq “ aeµt ` be´µt `1

2µ

ż 1

0e´µ|t´s|fpsqds, 0 ď t ď 1,

up0q “ a` b`1

2µ

ż 1

0e´µsfpsq ds “ 0,

up1q “ aeµ ` be´µ `e´µ

2µ

ż 1

0eµsfpsq ds “ 0,

where the complex numbers a “ apf, µq and b “ bpf, µq still have to bedetermined. The two boundary conditions above are equivalent to

ˆ

apf, µqbpf, µq

˙

“1

e´µ ´ eµ

ˆ

e´µ ´1´eµ 1

˙

˜

´ 12µ

ş10 e´µsfpsqds

´ e´µ

2µ

ş10 eµsfpsqds

¸

“1

2µpe´µ ´ eµq

˜

e´µş10pe

µs ´ e´µsqfpsq dsş10pe

µe´µs ´ e´µeµsqfpsqds

¸

.

We thus obtain λ P ρpAq and

Rpµ2, Aqfptq “ apf, µqeµt ` bpf, µqe´µt `1

2µ

ż 1

0e´µ|t´s|fpsqds

for all µ2 “ λ P CzR´, Reµ ą 0, f P X and t P r0, 1s.Fix φ P pπ2 , πq. Take λ P Σφ and thus µ P Σφ

2. Hence µ “ |µ| eiθ with

0 ď |θ| ă φ2 and Reµ “ |µ| cos θ ě |µ| cos φ2 . So we can estimate

Rpλ,Aqf8 ď |apf, µq| eReµ ` |bpf, µq|` ‖f‖82|µ|

suptPr0,1s

ż t

t´1e´Reµ|r| dr

5.2. Sectorial operators 88

ď‖f‖8

2|µ| |eµ ´ e´µ|

´

ż 1

0

`

eReµs ` e´Reµs˘

ds

`

ż 1

0peReµe´Reµs ` e´ReµeReµsq ds

¯

`‖f‖8|µ|Reµ

“‖f‖8

2 Reµ|µ||eµ ´ e´µ|`

peReµ ´ 1` 1´ e´Reµq

` eReµp1´ e´Reµq ` e´ReµpeReµ ´ 1q˘

`‖f‖8|µ|Reµ

ď

1cospφ2q

|µ|2‖f‖8

´

peReµ ´ e´Reµq ` peReµ ´ e´Reµq

2peReµ ´ e´Reµq` 1

¯

“

2cospφ2q

|λ|‖f‖8. l

Note that DpAq “ tu P Xˇ

ˇup0q “ up1q “ 0u is not equal to X for theabove ’Dirichlet-Laplacian’, cf. Example 1.19 in [Sc2].

Similarly one can show that the ‘Neumann-Laplacian’ A1u “ u2 with

DpA1q “ tu P C2pr0, 1sq

ˇ

ˇu1p0q “ u1p1q “ 0u

is sectorial for every angle φ ă π with on X “ Cpr0, 1sq. Moreover, itsspectrum is given by σpA1q “ σppA1q “ t´π

2k2ˇ

ˇ k P N0u with eigenfunctionsukptq “ cospkπtq. (See exercises.) Here DpA1q is dense in X.

Example 5.10. Let X “ LppRq, 1 ď p ă 8, and Au “ u1 for DpAq “W 1,ppRq. Then A is sectorial of any angle φ P p0, π2 q.

Proof. Example 3.37 says that σpAq “ iR and ‖Rpλ,Aq‖ ď 1Reλ for

Reλ ą 0. If φ P p0, π2 q and λ P Σφ, we have |Reλ| ě |λ| cosφ and hence

‖Rpλ,Aq‖ ď 1 cosφ

|λ|.

Because of its spectrum, A is not sectorial of angle φ “ π2 . l

Let 1 ă p ă 8, U Ď Rd be open and bounded with BU of class C2, andX “ LppRdq. The operators

A0 “ ∆, DpA0q “W 2,ppUq XW 1,p0 pUq,

A1 “ ∆, DpA1q “ tu PW2,ppUq

ˇ

ˇ Bνu “dÿ

j“1

νj tr Bju “ 0 on BUu,

are sectorial on X with angle φ ą π2. Here tr : W 1,ppUq Ñ LppBUq is thetrace operator and ν is the outer unit normal. There are variants for thespaces X “ L1pUq and X “ CpUq as well as for more general differentialoperators and boundary conditions. See e.g. Chapter 3 in [Lu] and also [Ta].To obtain these results, one has to rely on methods from partial differentialequations. One can however define the operators Aj in a more abstractway (in the ‘weak sense’) and show their sectoriality by purely functionalanalytic methods, cf. [Ou]. In this case, however, the domains are not knownexplicitly.

5.2. Sectorial operators 89

Let A be sectorial of angle φ P pπ2 , πq with constant K. Take any r ą 0and θ P pπ2 , φq. We define

Γ1 “ tλ “ γ1psq “ p´sqe´iθ

ˇ

ˇ ´8 ă s ď ´ru,

Γ2 “ tλ “ γ2pαq “ reiαˇ

ˇ ´ θ ď α ď θu,

Γ3 “ tλ “ γ3psq “ seiθˇ

ˇ r ď s ă 8u,

Γ “ Γpr, θq “ Γ1 Y Γ2 Y Γ3.

For t ą 0, we introduce the operator

etA “1

2πi

ż

ΓetλRpλ,Aqdλ “ lim

RÑ8

1

2πi

ż

ΓR

etλRpλ,Aq dλ, (5.5)

where ΓR “ ΓX Bp0, Rq for R ą r. We first have to show that the limit in(5.5) exists in BpXq.

Lemma 5.11. Under the above assumptions, the integral in (5.5) convergesabsolutely in BpXq and gives an operator etA P BpXq which does not dependon the choice of r ą 0 and θ P pπ2 , φq. Moreover, ‖etA‖ ď M for all t ą 0and a constant M “MpK, θq ą 0.

Proof. Since Rpλ,Aq ď K|λ| on Γ, we can estimate

ˇ

ˇ

ˇ

ˇ

ż

ΓR

‖etλRpλ,Aq‖ dλ

ˇ

ˇ

ˇ

ˇ

ď K

ż R

r

expptsRe e´iθq

|se´iθ||e´iθ| ds

`K

ż θ

´θ

expptrRe eiαq

|reiα||ireiα|dα

`K

ż R

r

expptsRe eiθq

|seiθ||eiθ|ds

ď K

ˆ

2

ż 8

r

ets cos θ

sds`

ż θ

´θetr cosα dα

˙

ď K

˜

2

ż 8

rt|cos θ|

e´σ

σp´t cos θq

dσ

´t cos θ` 2θetr

¸

“: Kcpr, t, θq,

for all R, t ą 0, where we substituted σ “ ´st cos θ. Thus the limit in (5.5)exists absolutely in BpXq by the majorant criterium, and ‖etA‖ ď Kcpr, t, θq.If we take r “ 1t, then cp1t, t, θq “: cpθq does not depend on t ą 0.

So it remains to check that the integral in (5.5) is independent of r ą 0and θ P pπ2 , φq. To this aim, we define Γ1 “ Γpr1, θ1q for some r1 ą 0 and θ1 P

pπ2 , φq, where we may assume that θ1 ě θ. We further set Γ1R “ Γ1XBp0, Rq

and choose R ą r, r1. Let C`R and C´R be the circle arcs from the endpointof ΓR to that of Γ1R in tImλ ą 0u and tImλ ă 0u, respectively. (If θ “ θ1,then C˘R contain just one point.) Then SR “ ΓR YC

`R Y p´Γ1Rq Y p´C

´R q is

a closed curve in the starshaped domain Σφ. So (5.1) shows thatż

SR

etλRpλ,Aq dλ “ 0.

5.2. Sectorial operators 90

We further estimate›

›

›

ż

C`R

etλRpλ,Aq dλ›

›

›ď

ż θ1

θetRRe eiα K

|Reiα||iReiα|dα ď Kpθ1 ´ θqetR cos θ Ñ 0,

as RÑ8, and analogously for C´R . So we conclude thatż

ΓetλRpλ,Aq dλ “ lim

RÑ8

ż

ΓR

etλRpλ,Aqdλ “ limRÑ8

ż

Γ1R

etλRpλ,Aq dλ

“

ż

Γ1etλRpλ,Aq dλ.

We next establish some of the fundamental properties of the operatorsetA. In view of these results one calls petAqtě0 the ‘holomorphic semigroupgenerated by A’. Actually, the theorem admits a converse. We refer toSection 2.1 of [Lu] for this and other facts.

Theorem 5.12. Let A be sectorial of angle φ ą π2 . Define etA as in (5.5)

for t ą 0, and set e0A “ I. Then the following assertions hold.(a) etAesA “ esAetA “ ept`sqA for all t, s ě 0.(b) The map t ÞÑ etA belongs to C1pp0,8q,BpXqq. Moreover, etAX Ď

DpAq, ddte

tA “ AetA and ‖AetA‖ ď Ct for a constant C ą 0 and all t ą 0.

We also have AetAx “ etAAx for all x P DpAq and t ě 0.

(c) Let x P X. Then etAx converges as tÑ 0 in X if and only if x P DpAq.In this case, etAx tends to x as tÑ 0.

Proof. (a) Let t, s ą 0. Take 0 ă r ă r1 and π2 ă θ1 ă θ ă φ. Set

Γ “ Γpr, θq and Γ1 “ Γpr1, θ1q. Using the resolvent equation and Fubini’stheorem, we compute

etAesA “1

p2πiq2

ż

Γetλ

ż

Γ1esµRpλ,AqRpµ,Aquddµdλ

“1

2πi

ż

ΓetλRpλ,Aq

1

2πi

ż

Γ1

esµ

µ´ λdµ dλ

`1

2πi

ż

Γ1esµRpµ,Aq

1

2πi

ż

Γ

etλ

λ´ µdλ dµ.

Fix λ P Γ and take R ą maxtr, r1, |λ|u. We define C 1R “ tz “ Reiαˇ

ˇ θ1 ďα ď 2π´θ1u and S1R “ Γ1RYC

1R. Since npS1R, λq “ 1, Cauchy’s formula (5.2)

yields1

2πi

ż

S1R

esµ

µ´ λdµ “ esλ.

As in Lemma 5.11, we further computeż

Γ1R

esµ

µ´ λdµ ÝÑ

ż

Γ1

esµ

µ´ λdµ and

ˇ

ˇ

ˇ

ˇ

ˇ

ż

C1R

esµ

µ´ λdµ

ˇ

ˇ

ˇ

ˇ

ˇ

ď 2πR supµPC1R

esReµ

|µ´ λ|ď esR cos θ1 2πR

R´ |λ|ÝÑ 0

as RÑ8. Consequently,

esλ “1

2πi

ż

Γ1

esµ

µ´ λdµ.

5.2. Sectorial operators 91

Closing ΓR with the circle arc CR “ tz “ Reiαˇ

ˇ θ ď α ď 2π ´ θu forsufficiently large R ą r, one verifies in the same way that

0 “

ż

Γ

eλt

λ´ µdλ.

We thus conclude that

etAesA “1

2πi

ż

ΓeλtesλRpλ,Aq dλ “ ept`sqA “ esAetA.

(b) Let x P X, t ą 0, ε ą 0, and R ą r. Observe that the Riemann sumsfor

ş

ΓRetλRpλ,Aqdλ converge in rDpAqs since λ ÞÑ Rpλ,Aq is continuous in

BpX, rDpAqsq. We thus obtain

A

ż

ΓR

eλtRpλ,Aq dλ “

ż

ΓR

etλARpλ,Aqdλ

“

ż

ΓR

eλtλRpλ,Aqdλ´

ż

ΓR

etλ dλ I. (5.6)

Take again CR “ tµ “ Reiαˇ

ˇ θ ď α ď 2π ´ θu. Using (5.1), one shows as inpart (a) thatˇ

ˇ

ˇ

ˇ

ż

ΓR

etλ dλ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

´

ż

CR

etλ dλ

ˇ

ˇ

ˇ

ˇ

ď 2πR supθďαď2π´θ

etR cosα ď 2πR eεR cos θ ÝÑ 0

as R Ñ 8, uniformly for t ě ε. Moreover, as in the proof of Lemma 5.11(with r “ 1t) we estimate

ˇ

ˇ

ˇ

ˇ

ż

ΓR

‖λetλRpλ,Aq‖ dλ

ˇ

ˇ

ˇ

ˇ

ď K´

2

ż 8

1t

s

sets cos θ ds`

ż θ

´θrecosαdα

¯

ď2K

t|cos θ|`

2eKθ

t“:

C 1

t.

Therefore, the right hand side of (5.6) converges toż

ΓλeλtRpλ,Aq dλ

as RÑ8. Since A is closed, it follows that etAX Ď DpAq and

AetA “1

2πi

ż

ΓλeλtRpλ,Aq dλ, ‖AetA‖ ď C 1

2πt

for all t ą 0. In a similar way one sees thatˇ

ˇ

ˇ

ˇ

ˇ

ż

ΓzΓR

λetλRpλ,Aqdλ

ˇ

ˇ

ˇ

ˇ

ˇ

ď 2K

ż 8

Rets cos θ ds ď

2K

ε|cos θ|eRε cos θ ÝÑ 0

as RÑ8, uniformly for t ě ε. As a result,ż

ΓR

λeλtRpλ,Aqdλ “d

dλ

ż

ΓR

eλtRpλ,Aq dλ

converges in BpXq uniformly for t ě ε, and so t ÞÑ etA P BpXq is continuouslydifferentiable for t ą 0 with d

dtetA “ AetA. For x P DpAq, we further obtain

AetAx “ limRÑ8

1

2πi

ż

ΓR

eλtRpλ,AqAx dλ “ etAAx.

5.2. Sectorial operators 92

(c) Let x P DpAq, R ą r, and t ą 0. As in part (a), Cauchy’s formula(5.2) yields

1

2πi

ż

Γ

eλt

λdλ “ lim

RÑ8

1

2πi

ż

ΓR

eλt

λ´ 0dλ “ 1

Observing that λRpλ,Aqx´ x “ Rpλ,AqAx, we conclude that

etAx´ x “1

2πi

ż

Γeλt

´

Rpλ,Aq ´1

λ

¯

x dλ “1

2πi

ż

Γ

eλt

λRpλ,AqAx dλ.

Since the integrand is bounded by c|λ|2 on Γ for all t P p0, 1s, Lebesgue’s

convergence theorem implies the existence of the limit

limtÑ0

etAx´ x “1

2πi

ż

Γ

1

λRpλ,AqAx dλ “: z.

Let KR “ tReiαˇ

ˇ ´ θ ď α ď θu. Cauchy’s theorem (5.1) shows thatż

ΓRYp´KRq

1

λRpλ,AqAx dλ “ 0.

Since also›

›

›

›

ż

´KR

1

λRpλ,AqAx dλ

›

›

›

›

ď2πRK

R2‖Ax‖ ÝÑ 0

as RÑ 8, we arrive at z “ 0. Because of the uniform boundedness of etA,it follows that etAxÑ x as tÑ 0 for all x P DpAq.

Conversely, if etAx Ñ y as t Ñ 0, then y P DpAq by part (b). Moreover,Rp1, AqetAx “ etARp1, Aqx tends to Rp1, Aqx as t Ñ 0, since Rp1, Aqx P

DpAq. We thus obtain Rp1, Aqy “ Rp1, Aqx, and so x “ y P DpAq.

Remark 5.13. Let A´ ωI “ Aω be sectorial of angle greater than π2 for

some ω P R. We then compute

eωtetAω “1

2πi

ż

Γetpλ`ωqRpλ` ω,Aq dλ “

1

2πi

ż

ω`ΓeµtRpµ,Aq dµ “: etA

for all t ą 0. It is easy to see that etA “ eωtetAω has the analogous propertiesas in the case ω “ 0. ♦

We can now easily solve the evolution equation (5.7) governed by a sec-torial operator A with angle φ ą π2. One calls such problems ‘of parabolictype’ since diffusion problems are typical applications, see Example 5.15.In contrast to the Schrodinger equation in Example 4.19 we can allow forinitial values in DpAq.

Corollary 5.14. Let A be sectorial of angle φ ą π2 and let u0 P DpAq.

Then uptq “ etAu0, t ě 0, is the unique solution in C1pp0,8q, Xq XCpp0,8q, rDpAqsq X CpR`, Xq of the initial value problem

u1ptq “ Auptq, t ą 0, up0q “ u0. (5.7)

Proof. Existence follows from Theorem 5.12 and Remark 5.13. Let vbe another solution of (5.7). Let 0 ă ε ď s ď t´ ε ă t. Theorem 5.12 thenimplies that

ddse

pt´sqAvpsq “ ´ept´sqAAvpsq ` ept´sqAv1psq “ 0.

5.2. Sectorial operators 93

As in Example 4.19, this fact yields ept´εqAvpεq “ eεAvpt´εq. Letting εÑ 0,one obtains that etAu0 “ vptq since τ ÞÑ eτAx is continuous for τ ě 0 and

each x P DpAq.

We only give one of the possible examples.

Example 5.15. Let X “ Cpr0, 1sq and Aϕ “ ϕ2 with DpAq “ tϕ P

C2pr0, 1sqˇ

ˇϕp0q “ ϕp1q “ 0u. Let u0 P C0p0, 1q “ DpAq. Then the function

uptq “ etAu0, t ě 0, belongs to

CpR`, Cpr0, 1sqq X Cp0,8q, C2pr0, 1sqq X C1pp0,8q, Cpr0, 1sqq

and uniquely solves the partial differential equation

Btupt, xq “ Bxxupt, xq, t ą 0, x P r0, 1s,

upt, 0q “ upt, 1q “ 0, t ą 0, (5.8)

up0, xq “ u0pxq, x P r0, 1s. ♦

The formula (5.5) and Theorem 5.12 allow to establish a theory for para-bolic problems which is similar to the case of ordinary differential equations,see e.g. [He] and [Lu]. We only add a basic result on the longterm behaviorthat extends Example 5.4. The next theorem can be applied to (5.8) thanksto Example 5.9.

Theorem 5.16. Let A be sectorial of angle φ ą π2 and spAq “suptReλ

ˇ

ˇλ P σpAqu ă ´δ ă 0. Then there is a constant N ě 1 such

that etA ď Ne´δt for all t ě 0.2

Proof. The assumptions imply that A´δ “ A` δI is sectorial of someangle ψ P pπ2, φq. Take Γ “ Γpr, θq with r ą 0 and θ P pπ2, ψq. Re-mark 5.13 then yields that etA “ e´δtetA´δ , where however etA is defined bythe curve integral (5.5) on the shifted path Γ1 :“ ´δ`Γ. Lemma 5.11 showsthat etA´δ is uniformly bounded for t ě 0. It thus remains to verify that

ż

´δ`ΓetµRpµ,Aq dµ “

ż

ΓetλRpλ,Aq dλ, t ą 0. (5.9)

To this end, let R ą r and S˘R be the horizontal line segments connectingthe end points on ΓR and Γ1R in tImλ ą 0u and tImλ ă 0u, respectively.Let CR “ ΓR Y S`R Y p´Γ1q Y p´S´R q. This path is contained in ρpAq andnpCR, zq “ 0 for all z P σpAq. Cauchy’s theorem (5.1) now implies

ż

CR

etλRpλ,Aqdλ “ 0.

Observe that the segments S˘R have fixed length δ and that Reλ ď R cos θ ă

0 and |λ| ě R for all λ P S˘R . Because of θ ă φ, the sets S˘R belong to Σφ

for all sufficiently large R. We can thus estimate›

›

›

ż

S˘R

etλRpλ,Aq dλ›

›

›ďδK

ReRt cos θ.

Since the right-hand side vanishes as RÑ8, we have shown (5.9).

2Not shown in the lectures.

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