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

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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)

|z|

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)

D(Bp) := Dp |x| ,

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.

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