On the Decay of Bounded Semigroup on the Domain of its Generator

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Vietnam J. Maih. (2015) 43:207-213 DOI 10.1007/S10013-014-0093-Z

On the Decay of Bounded Semigroup on the Domain of its Generator

Grigory M. Sklyar

Received 14November 2013/Accepted: 13 April 2014/Pubhshed online; 18 September 2014

Abstract For a bounded Co-semigroup on a Banach space X, we prove the following statement: the rate of decay of the semigroup on the domain of its generator is bounded by some decreasing function if and only if the spectmm of die semigroup does not contain any pure imaginary points. Our approach is based on the analysis of a special semigroup on the space of bounded linear operators . ^ ( X , X).

Keywords Bounded Co-semigroup • Rate of decay Strong asymptotic stability • Spectrum • Domain of generator

Mathematics Subject Classifications (2010) 34E99 • 47D03 47D06

We begin with recalling the following remarkable result ' on asymptotic stability.

Theorem I Ui A be the generator ofa bounded Co-semigroup {e'^'\i>o on a Banach space X and let intersection of the spectrum oiA) with the imaginary axis aiA) Ci ((K) be al most countable Then the semigroup (^'*'|/>o is strongly asymptolicaliy stable ii.e., lim,_.+^ ile'^'^cjl = 0 / o r all x e X) if and only if the adjoint operator A* has no pure imaginary eigenvalues.

This fact was proved by Sklyar and Shirman in 1982 [ 11] for the case of bounded operator A. The methodof treating of this problem given in [II] was picked up by Lyubich and

'We give an equivalent formulation ofihe resull.

Dedicated lo my friend. Professor Nguyen EGioa Son CM. Sklyar (S)

Instimte of Mathematics. University of Szczecin, Wielkopolska 15. 70-451 Szczecin. Poland e-mail. [email protected]

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Vu Phong [9] who extended in 1988 the resuh to the general case Independently, in 1988, Theorem 1 was obtained by Arendt and Batty [1] who used some different approach.

Let us note that the norm of the semigroup from Theorem 1 may not lend to zero (diis occurs if the unifoim growth bound OM, = lim,-.+c« log \\e" \\/t = 0). In this case, one can see that among die solutions of the abstract Cauchy problem

i^Axit).

jr(0)-JCo, ( > 0 .

Xo e X, (1)

given by xit) = e^'jco. l >O.XQe X, there are some tending to zero arbitrarily slow. At the same time. Batty [2, 3] and Vu Phong [10] proved the following result (formulation is taken from [5])

Theorem 2 Ul fe'^'}f>0 be a bounded Co-semigroup on a Banach space X wilh generator A and

<j(A)n(/K) = 0 (2) Then

In other words, all the classical solutions of problem (1) (i.e., those for J:O € !^(A)) tend to zero no slower than a certain decreasing function git) = \\Til)A~^\\, i.e.,

xit) = e^'xo = &igit)). t^oo, xoe 3!iA).

The proof of Theorem 2 given in [2, 3] is based on the application of some results and methods from [7, 8]. In 2008, Batty and Duyckaerts considered [4] a question of necessity of condiHon (2) in Theorem 2 for the decay of the norm of the semigroup on the domain ^ ( A ) . Their proof is based on careful estimations of some special characteristics of semigroup norm The main result of [4] in equivalent form is as follows.

Theorem 3 A hounded Co-semigroup [e^'],>Q on a Banach space X with generator A satisfies the relation

| € ' " ( A - X / ) - ' | ^ 0 . t ^ H-oo, X^aiA).

if and only if condition (2) holds.

The main goal of the present note is to present another perspective on the subject discussed above, namely we show that Theorem 3 can be proved as a straightforward con- clusion of Theorem 1 if we introduce a special semigroup on a space of bounded linear operators, and apply Theorem 1 to this semigroup. Moreover, we observe that our proof of

"only if" part of Theorem 3 does not exploit the assumption of boundedness of the semigroup (see Remark 1). Further development of the ideas of this approach will be the topic of a forthcoming paper

2 The Space and the Semigroup

Let A be an infinitesimal operator in a Banach space X and {e^'],^o be the semigroup generated by A. Let us consider the space i f f X. X) of linear-bounded operators from X to

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On the Decay of Bounded Semigroup on the Domain of ils Generator X and its subspace Y c Sf(X, X) defined by

¥ = {DR>.iA). D € L(X, X)]. X i CT(A).

where R\iA) = (A—A/)~' and Q denotes the closure of the linear set Q taken with respect to the norm of ^{X, X). It is clear that Y does not depend on X. Next, we introduce an operator semigroup {r(r)|,>o on the space Y given by the formula

T{j)B = Be^'. B EY.

Proposition 1 {7"(f)},>o is a Co-semigroup.

Proof Let BQ € Y. For any e > 0, we choose an operator B ^ DR-^A), D € .if (X, X) such that IJfi - Soil < f Then we get the following esUmate for / € [0, toV

\\fU)Bo - Boll - ||So(^'" - / ) | | < | | e ( ^ ' " - / ) | | + | ( 1 + Mo). (3) where Mo is max,g[o,;o] ||^'" [|. Using the form of B we obtain

JBie'^' - / ) | | < IIDIJ ||fix(A)e'" - RxiA)j . For any x e X we have

(^R-JA)e'" - R;.iA))x^ f Z " " ^ ' (e'^' + XR).iA)) xdz (4) and hence

\\R)^iA)e'" -RAA)\sMit(,.

where My ^ max,,i6[o,,„| | e ^ " - ' ' | | | e ' ^ ' -I- XRx{A)\\. Then from (3) we infer

Now choosing to > 0 such a small number that

||D||Mi/o < ^ . Wo < 1 we get

| f ( ( ) e o - 5 o | | < £

that means the strong continuity of the semigroup Tir). D Denote by A the generator of the semigroup [7"(r}}f>o Let ^I C >" be the set

Yl ={BR>,(A), Be Y}. X^aiA).

One can see that I'l does not depend on the choice of JL.

Proposition 2 Operator A is defined on the set Y\ and given there b^ the formula AiBRxiA)) = Bil+XRxiA)), BeY.

Proof We need to prove the relation

lim - [fir)iBRUA)) - fiftj.(A)] = BU+XRxiA)). BeY. (5)

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210 G.M Sklyar where limit on the tefl hand side of (5) is regarded m the sense of the space Y. First, we recall that

fit)iBR^iA)) - BKi(A) - BRxiA) (e^' - / ) . (6) Using (4) for any x e X, we have

BRxiA){e'^' -!)x- = B j e^'-'''^ (e^' +XRiXA))xdT

= [ e^^'-'UfiT)B + kBR)iA))xdT.

Jo

Since the function Tit)B is continuous in the norm of .if (X. X) (see Proposition 1) dien / e^-^'-'^f(T)BxdT = [ e^^'-'^fiz)Bdzx. x E X, Jo Jo and hence we obtain the following operator equality

ixiA) (^e^' - 0 " / ^^""'^ f^'"^*^ + >^BRxiA)) dv. (7)

- [f(i)iBR;^(Ay) - BRxiA)] - BU + XR>XA)) =^—J' Pix)dT + (e^' - l) B(l + kRyXA)), where

f (r) - e-^' {fiT)B + XBRxiA)) - Bil + XR^iA))

The funciion Fix) is continuous in the norm of i f ( X . X) and F(0) - 0. Hence

\\J^F{r)dT\\ < f - C„ where C, = max,g[o,,j ||f (T)|| ^ 0, / - ^ 0, this yields

V / o f^('^)dT - * 0, / -> 0, what implies (5) The proof is complete. D

3 Spectral Properties

We prove two lemmas on the spectrum of the operator A defined in die previous section.

Lemma 1 The domam ^ ( A ) of A is exactly the .set F, and the spectrum aiA) verifies the

inclusion ^ al.A) C<J(A).

Proof We already proved Propos.fon 2 Ihat >-, c »(A). So we need lo show (he implication!

if»S« y, iheaSfSiA).

To this end. we consider an arbitrary >. i <T(A} and the operator A - i ; We observe that is ine same. BHxiA) e Fj, then {Proposition 2)

(A - kl) BR^iA) = Bil + kRxiA)) - XBR^iA) = B.

So the mapping

BR,.IA) t i ' B

IS one-to one. Besides, this operator has a bounded inverse given by

{A-uy' B = BRx(A). BsY.

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On ihe Decay of Bounded Semigroup on the Domam of ils Generaior This means that

1. XiaiA);

2. S ? ( A ) ^ ( A - A / ) - ' Y = Y\

The lemma is proved n Lemma 2 Ut p e C fee a boundary point of the spectrum fr(A). Then ii e (7(A) and

moreover p is an eingenvalue ofihe adjoint operator A".

Proof For simplicity let us put n ^ 0. Then from the theorem of boundary point of spectrum ([6, 12]), there exists a sequence [x^] C ®(A) such that

1. Ik„|| = I, « e N , 2. JIAxJI^O, n^oo.

LeiA^cr(A).ThenA/?j.(A).v„ = flj^(A)Ax„ ^ 0. H ^ oc. This yields

\\Rx(A)x„\\ = WîARîA) - I)x„\\ ^ i - . nôo. (8) II-^ ll 1^1

Let Y2 be the image of A.

^ 2 - A ^ ( A ) - A K ] . If D e Y2 then (see Proposition 2)

D = B(/ -I- kRxiA)) = BAR),iA).

Hence

Dx„ = BRxiA)Ax„ -^0. n^ 00. (9) Relations (8), (9) imply that

J„fJ|«.(A,-Z,||>±>0.

This means that R}.iA) f Y2 and there exists a nonzero funcnonal f eY* such that / ( Z ) ) ^ 0 . DEY2.

This means that

A * / = 0

and the lemma is proved. D

4 Proof of Theorem 3

Let us assume now that the semigroup {e'^'],->Q with generator A is bounded. Let us prove Theorem 3.

Proof Sufficiency. First, we observe that the semigroup, given by the law fit)B^Be^'. BEY.

is obviously also bounded. Due to Lemma 1, we have fT(A)n(/R) C(T(A)n(;lR),

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G M Sklyar

where A is the generator of i?(r)|,>.i. So if conjiiuon (2) holds, then die set cr(A) n UR) is empty. Applymg now Theorem 1 for operator A and semigroup Til), we obtain

fit)B -^0, BeY.

In particular, if B ^ RkiA) we obtain the sufficiency.

Necessity. Assume that condition (2) does noi hold, then there exists a boundary point p of CT(A) witti R e n > 0 Due to Lemma 2, we conclude that p is an eigenvalue of A*, i.e., diere exists f e Y ' . / /= 0, such that

f'it)f = e'"j.

Let Do e >' be such that /(Z>o) ?t 0. It is clear that Do can be chosen m the form Do ^ Bfi,,(A). Then we have

{fm)f)iDo) = f(fil)Do)^e^'fiDo).

Since \e^'\ > I and / ( D o ) 7^ 0 this implies ihal

f(f)Do = B e ' " RkiA) / * 0' ' ^ + 0 ^ and therefore

/'flx(A) A 0, I -^ +00.

This completes the prooL ^ Remark I Let us observe thai the proof of necessity in Theorem 3 does not use the

boundedness of semigroup e^'. That means that the following statement holds.

If Co-semigroup (e'"|r>o (not necessarily bounded) on a Banach space X with generator A satisfies the relation

L ' " { A - i / ) - ' | | - > 0 . f ^ +CO, XiuiA), ihen

aiA) c \X:K^k < 0 | .

Acknowledgments The author ihanks the anonymous referees for valuable remarks and recommendations which have improved the exposition of the article

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use. dislribulion, and reproduction m any medium, provided the original aulhor{s) and the source are credited

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