A class of relatively bounded perturbations for generators of bounded analytic semigroups in Banach...

J. Math. Anal. Appl. 416 (2014) 855–861

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

A class of relatively bounded perturbations for generatorsof bounded analytic semigroups in Banach spaces

Motohiro Sobajima a,b

a Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, 162-8601,Tokyo, Japanb Dipartimento di Matematica “Ennio De Giorgi”, Università del Salento, C.P. 193, 73100, Lecce, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 November 2013Available online 18 March 2014Submitted by E. Saksman

Keywords:Relatively bounded perturbationsAnalytic semigroupsInverse square potentials in Lp

Generation of analytic semigroups on Banach space X by −(A + kB) is shown,where A is a negative generator of a bounded analytic semigroup on X, B is a closedoperator in X belonging to a class related to A and k ∈ C is a parameter satisfyingRe k > c for some c ∈ R. The proof may be regarded as a modification of theperturbation argument established by Okazawa [8]. As applications, the generationof non-contractive analytic semigroups by Schrödinger operators with inverse squarepotentials in Lp(RN ) is discussed.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction and result

Let X be a Banach space over C and let A and B be linear operators in X with domains D(A) and D(B),respectively. In this paper we discuss relatively bounded perturbations for generators of bounded analyticsemigroups on X. Namely, we consider the generation of analytic semigroup on X by the operator

−(A + kB) with domain D(A + kB) := D(A)

for k ∈ C \ {0} when −A generates an analytic semigroup on X and B is relatively bounded with respectto A.

The theory of relatively bounded perturbations for generators of analytic semigroups is well-known (seee.g., Kato [6, Section IX.2], Engel and Nagel [4, Section III.2], Desch and Schappacher [3] and Gershtein [5])for small or compact perturbations.

On the other hand, Okazawa showed in [8] that A+B is m-accretive in X, that is, −(A+B) generates acontraction semigroup on X if A and B are m-accretive in X and for every ε > 0, there exists f ∈ F (Bεu)such that

Re(Au, f)X,X′ � −c‖u‖2 − a‖u‖ · ‖Bεu‖ − b‖Bεu‖2, u ∈ D(A) (1.1)

E-mail address: msobajima1984@gmail.com.

http://dx.doi.org/10.1016/j.jmaa.2014.03.0310022-247X/© 2014 Elsevier Inc. All rights reserved.

856 M. Sobajima / J. Math. Anal. Appl. 416 (2014) 855–861

for some constants a, c � 0 and b < 1 (independent of ε), where F is the (possibly multi-valued) dualitymap from X to its dual space X ′:

F (v) :={f ∈ X ′; ‖f‖ = ‖v‖ and (v, f)X,X′ = ‖v‖2},

Bε is the Yosida approximation of the m-accretive operator B, and (w, g)X,X′ is the pairing betweenw ∈ X and g ∈ X ′. Namely, Okazawa succeeded in proving the generation of a contraction semigroup by−(A + B) without a smallness assumption on B. Moreover, the closedness of A + B is also given in [8]under condition (1.1). It is worth noticing that the result can be easily applied to some partial differentialoperators by using integration by parts.

Recently, Maeda and Okazawa [7] obtained that −(A+kB) generates an analytic contraction semigroupif −A generates an analytic contraction semigroup, Re k > 0 and B satisfies that for every u ∈ X andf ∈ F (Bεu),

Im(u, f)X,X′ = 0, Re(u, f)X,X′ � 0. (1.2)

The first purpose of this paper is to present a simple perturbation result for analytic semigroups underrelatively bounded, not small, perturbation (see Theorem 1.1). The second is to discuss the generation ofanalytic semigroups by Schrödinger operators with inverse square potentials −Δ+λ|x|−2 with λ > −(N−2

2 )2in Lp(RN ) (N � 3, 1 < p < ∞) (see Theorem 3.1).

Now we are in a position to state the main result. For simplicity, we assume that the duality map F issingle-valued and denote the element of F (w) by the same notation (for the case where F is multi-valued,see also Remark 1.2).

Theorem 1.1. Let A and B be densely defined operators in X. Assume that

i) −A generates a bounded analytic semigroup on X;ii) D(A) ⊂ D(B) and there exists β0 > 0 such that for every u ∈ D(A),

Re(Au,F (Bu)

)X,X′ � β0‖Bu‖2, (1.3)(

u, F (Bu))X,X′ � 0. (1.4)

Then for every k ∈ C satisfying Re k > −β0, −(A+kB) with domain D(A) also generates a bounded analyticsemigroup on X. Moreover, if A has a compact resolvent, then A + kB also has a compact resolvent.

Assume further that X is a Banach lattice, A has positive resolvent and B is also positive. Then A+ kB

is also positive resolvent for every k ∈ (−β0, 0).

Remark 1.1. Condition (1.3) is stronger than the relative boundedness with respect to A and also (1.1). Infact, from (1.3) we have

‖Bu‖ � 1β0

‖Au‖

for u ∈ D(A). Condition (1.4) is similar to (1.2) which essentially appears in [7, Lemma 3.2].

Remark 1.2. If the duality map F is multi-valued, then we replace conditions (1.3) and (1.4) with thefollowing one: for every u ∈ D(A), there exists f ∈ F (Bu) such that

Re(Au, f)X,X′ � β0‖Bu‖2, (1.5)

(u, f)X,X′ � 0. (1.6)

M. Sobajima / J. Math. Anal. Appl. 416 (2014) 855–861 857

It is worth noticing that, in general, the existence of f ∈ F (Bu) satisfying (1.5) and (1.6) does not imply (1.5)and (1.6) for every f ∈ F (Bu). In fact, we consider X = C

2 endowed with the norm ‖(x1, x2)‖ = |x1|+ |x2|.Let A(x1, x2) = (x1, x2) and B(x1, x2) = (x1, 0). In this case, we see that F (B(x1, x2)) = {(x1, z) ∈ C

2;z ∈ C, |z| � |x1|}. Hence we can verify (1.5) and (1.6) with β0 = 1 for every (x1, x2) ∈ C

2 by choosingz = 0. However, (1.5) and (1.6) fail if u = (1, 2) and f = (1,−1).

This paper is organized as follows. Theorem 1.1 is proved in Section 2. To illustrate it, we discuss thegeneration result for Schrödinger operators with inverse square potentials −Δ + λ|x|−2 in Lp(RN ) (N � 3,1 < p < ∞) in Section 3.

2. Proof of Theorem 1.1

For the proof of Theorem 1.1, we need the following lemma.

Lemma 2.1. Set C+ := {z ∈ C; z �= 0,Re z � 0}. Assume that A and B satisfy (1.3) and (1.4). Then forevery z ∈ C+, Reλ > −β0 and u ∈ D(A),

‖Bu‖ � 1Reλ + β0

‖zu + Au + λBu‖. (2.1)

Proof. Let u ∈ D(A). Then from (1.4) and (1.3) we obtain

Re(zu + Au + λBu, F (Bu)

)X,X′ = (Re z)

(u, F (Bu)

)X,X′ + Re

(Au,F (Bu)

)X,X′ + (Reλ)‖Bu‖2

� (Reλ + β0)‖Bu‖2.

Thus we have

(Reλ + β0)‖Bu‖2 � ‖zu + Au + λBu‖‖Bu‖,

and hence noting that Reλ + β0 > 0, we obtain (2.1). �Proof of Theorem 1.1. By [4, Section II.4.a], it suffices to prove that there exists a constant M � 1 suchthat

C+ ⊂ ρ(−(A + kB)

)(2.2)

and for every z ∈ C+,

∥∥(z + A + kB)−1∥∥L(X) �

M

|z| . (2.3)

Since −A is a generator of a bounded analytic semigroup on X, we have that

C+ ⊂ ρ(−A),

and there exists a constant M0 � 1 such that for every z ∈ C+,

∥∥(z + A)−1∥∥L(X) �

M0. (2.4)

858 M. Sobajima / J. Math. Anal. Appl. 416 (2014) 855–861

First fix k ∈ C satisfying |k| < β0 and let z ∈ C+. Then Using Lemma 2.1 with λ = 0, we have

∥∥kB(z + A)−1∥∥ � |k|β0

< 1.

Thus we see that 1 + kB(z + A)−1 is invertible and

∥∥(1 + kB(z + A)−1)−1∥∥ � β0

β0 − |k| .

And hence noting that

z + A + kB =[1 + kB(z + A)−1](z + A),

from (2.4) we obtain z ∈ ρ(−(A + kB)) and

∥∥(z + A + kB)−1∥∥ =∥∥(z + A)−1(1 + kB(z + A)−1)−1∥∥

� M0β0

β0 − |k| ·1|z| .

Next we iterate the same argument with A replaced with A+ (nβ0/2)B for n ∈ N. In this case we applyLemma 2.1 with λ = nβ0/2. Then we have (2.2) and (2.3) for k ∈ C satisfying |k − nβ0/2| < (1 + n/2)β0.Thus we obtain the generation of analytic semigroup on X for every k ∈ C satisfying Re k > −β0.

Now we assume that (z + A)−1 is compact for z ∈ C+. If |k| < β0, then (z + A + kB)−1 can be writtenby

(z + A + kB)−1 = (z + A)−1(1 + kB(z + A)−1)−1.

Hence we see that (z + A + kB)−1 is also compact. The case Re k > −β0 is treated as in the proof ofgeneration result.

Finally, we assume that X is a Banach lattice, and (1 +A)−1 and B are positive in X. Let k ∈ (−β0, 0).Note that

(1 + kB(1 + A)−1)−1 =

∞∑n=0

(−kB(1 + A)−1)n.

Since B(1 + A)−1 is positive, (1 + kB(1 + A)−1)−1 is also positive, and hence (1 + A + kB)−1 is alsopositive. �3. Schrödinger operators with inverse-square potentials in Lp

In this section we discuss generation results for Schrödinger operators as an application of Theorem 1.1.We consider Schrödinger operators with inverse square potentials in Lp(RN ) (N � 3, 1 < p < ∞):

⎧⎪⎨⎪⎩

Spu := −Δu + λ

|x|2u,

D(Sp) := W 2,p(R

N)∩Dp

(|x|−2)

for λ ∈ R satisfying λ > −(N−22 )2, where W 2,p(RN ) is the usual Sobolev space and Dp(|x|−2) := {w ∈

Lp(RN ); |x|−2w ∈ Lp(RN )} is the maximal domain of the multiplication operator |x|−2 in Lp(RN ).

M. Sobajima / J. Math. Anal. Appl. 416 (2014) 855–861 859

If λ < −(N−22 )2, then Baras and Goldstein [1] proved that no restriction of Δ − λ|x|−2 (in the sense of

distributions) generates a positive semigroup on Lp(RN ) for any 1 � p � ∞.The selfadjointness of S2 in L2(RN ) is described in Reed and Simon [10, Theorem X.11] if λ >

−(N−22 )2 + 1. The m-accretivity and m-sectoriality of Sp in Lp(RN ) are studied in Okazawa [9]. More

precisely, Okazawa proved in [9] that if p � N/2, then −Sp generates an analytic semigroup on Lp(RN )under

λ > β(p) := − (p− 1)N(N − 2p)p2 (� 0). (3.1)

On the other hand, if 1 < p < N/2, then the domain is characterized as D(Sp) = W 2,p(RN ) by using theRellich inequality

∥∥∥∥ u

|x|2

∥∥∥∥Lp(RN )

�(−β(p)

)−1‖Δu‖Lp(RN ), u ∈ W 2,p(R

N)

(see e.g., [9, Lemma 3.8] and [2, Theorem 12]) and −Sp generates an analytic semigroup on Lp(RN ) underboth conditions

λ > β(p) and λ � − (p− 1)(N − 2)2

p2 ;

note that condition λ � − (p−1)(N−2)2p2 is the necessary and sufficient condition for the accretivity of Sp (see

e.g., [9, Theorem 2.4]). In the case where 2(1 − 1/N) � p < ∞, the lower bound of λ in (3.1) is optimal inthe sense of generation of analytic semigroups on Lp(RN ).

The following theorem gives the precise lower bound of λ even if 1 < p < 2(1 − 1/N).

Theorem 3.1. Let N � 3 and 1 < p < ∞. If

λ > β(p) = − (p− 1)N(N − 2p)p2 ,

then −Sp generates a bounded analytic semigroup on Lp(RN ).

Remark 3.1. In particular, if 1 < p < 2(1 − 1/N) and

β(p) < λ < − (p− 1)(N − 2)2

p2 ,

then Sp is not accretive in Lp(RN ). This yields that the analytic semigroup generated by −Sp is notcontractive in Lp(RN ).

Proof of Theorem 3.1. We introduce the operators Ap and Bp respectively defined as

⎧⎪⎨⎪⎩

Apu := −Δu + λ0

|x|2u,

D(Ap) := W 2,p(R

N)∩Dp

(|x|−2),⎧⎨

⎩Bpu := u

|x|2 , ( −2)

860 M. Sobajima / J. Math. Anal. Appl. 416 (2014) 855–861

where

λ0 := 1 + max{0, β(p)

}.

Then by virtue of [9, Corollary 4.2], we see that Ap is m-sectorial in Lp(RN ). In other words, −Ap generatesan analytic contraction semigroup on Lp(RN ). Noting that Sp = Ap + (λ − λ0)Bp, we apply Theorem 1.1to Ap and Bp with k = λ− λ0.

Using [9, Lemma 3.5] (see also [7, Lemma 5.3]), we see that for every ε > 0 and v ∈ W 2,p(RN ),

(−Δv, F

(v

|x|2 + ε

))Lp,Lp′

� −β(p)∥∥∥∥ v

|x|2 + ε

∥∥∥∥2

Lp

,

where F (w) := ‖w‖2−pLp |w|p−2w is the duality map from Lp(RN ) to Lp′(RN ) and (|x|2 + ε)−1 is the Yosida

approximation of Bp = |x|−2. Choosing v ∈ D(Ap) and letting ε ↓ 0, we have

Re(−Δv, F (Bpv)

)Lp,Lp′ � −β(p)‖Bpv‖2

Lp ,

and hence we obtain (1.3):

Re(Apv, F (Bpv)

)Lp,Lp′ = Re

(−Δv, F (Bpv)

)Lp,Lp′ + λ0‖Bpv‖2

Lp

�(λ0 − β(p)

)‖Bpv‖2

Lp .

On the other hand, (1.4) is obvious: for every v ∈ D(Bp),

(v, F (Bpv)

)Lp,Lp′ = ‖Bpv‖2−p

Lp

∫RN

|v|p|x|2(p−1) dx � 0.

Applying Theorem 1.1 to Ap and Bp with k = λ−λ0 and β0 = λ0−β(p), we have that for every Reλ > β(p),the operator −Sp = −(Ap + (λ− λ0)Bp) generates a bounded analytic semigroup on Lp(RN ). �

Finally, we state a special result for a particular value of p. If N � 3 and p = 2NN+2 , then β(p) = −(N−2

2 )2.Therefore we have that the following proposition directly follows from Theorem 3.1.

Proposition 3.2. Let N � 3 and p = 2NN+2 . If λ ∈ R satisfies λ > −(N−2

2 )2, then −Sp with domain W 2,p(RN )generates a bounded analytic semigroup on Lp(RN ).

Acknowledgment

The author would like to thank Professor Giorgio Metafune for giving valuable comments for the refine-ment of this paper.

References

[1] P. Baras, J. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984) 121–139.[2] E.B. Davies, A.M. Hinz, Explicit constants for Rellich inequalities in Lp(Ω), Math. Z. 227 (1998) 511–523.[3] W. Desch, W. Schappacher, Some perturbation results for analytic semigroups, Math. Ann. 281 (1988) 157–162.[4] K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., vol. 194, Springer-

Verlag, New York, 2000.[5] L.M. Gershtein, Perturbation of the generating operator of an analytic semigroup, Funktsional. Anal. i Prilozhen. 22 (1988)

62–63; translation in Funct. Anal. Appl. 22 (1988) 51–52.

M. Sobajima / J. Math. Anal. Appl. 416 (2014) 855–861 861

[6] T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss., vol. 132, Springer-Verlag New York, Inc.,New York, 1966.

[7] Y. Maeda, N. Okazawa, Holomorphic families of Schrödinger operators in Lp, SUT J. Math. 47 (2011) 185–216.[8] N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan 34 (1982) 677–701.[9] N. Okazawa, Lp-theory of Schrödinger operators with strongly singular potentials, Jpn. J. Math. 22 (1996) 199–239.

[10] M. Reed, B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press,New York, London, 1975.