On the stability of strongly continuous semigroups of positive operators on <span class="mathjax-formula">$L^2 (\mu )$</span>

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

G. G ^REINER R. N ^AGEL

On the stability of strongly continuous semigroups of positive operators on L

²

(µ)

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4

^e

série, tome 10, n

^o

2 (1983), p. 257-262

<http://www.numdam.org/item?id=ASNSP_1983_4_10_2_257_0>

© Scuola Normale Superiore, Pisa, 1983, tous droits réservés.

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Stability Strongly Semigroups

of Positive Operators

^on

^L2(03BC).

G. GREINER - R. NAGEL

(*)

One of the fundamental

problems

ⁱⁿ infinite-dimensional linear

stability theory

^{is to} decide whether

for the

generator

^{A of a}

strongly

continuous

semigroup {T(t)}t_,o

^{of bounded}

linear

operators

^{on a} Banach space

(see [9]). Here,

denotes the

spectral

bound of A while

is the

growth

^{bound of}

~T(t)~t~o.

The coincidence of the

spectral

^and

growth

bounds means that

stability

^{of the}

semigroup depends

^on the location of the

spectrum

^{of the}

generator.

^More

precisely,

y suppose that

(*)

^{is true.}

Then

s(A)

⁰

implies lim

_t--

11 T(t) ==

^0.

We refer to

[7]

^{for a}

complete

discussion of

(*),

but recall that

« s (A )

⁼ coo » does not hold in

general,

neither for

semigroups

^{on Hilbert}

spaces

(see ^[10]

^or

^[2],

^th.

2.17)

^{nor for}

positive semigroups

^{on Banach}

lattices

(see [4]).

^{In this note we} combine the Hilbert space and the

(*)

Questo lavoro 6 stato

completato

^{durante un}

soggiorno

del secondo Autore presso l’Universith di Bari in

qualita

^di

professore

visitatore.

Quest’ultimo

coglie

l’occasione _per

ringraziare

^il Comitato delle Scienze Mate- matiche del C.N.R. _per aver permesso questo

soggiorno

^e l’Istituto di Analisi Mate- matica dell’Università di Bari _per la cortese

accoglienza.

Pervenuto alla Redazione il 10

Giugno

¹⁹⁸² ed in forma definitiva il 5 Feb- braio 1983.

258

order structure and show that

s(A)

⁼ for every

strongly

^continuous

semigroup

^on

consisting

^of

positive operators,

^{i.e. such that}

T(t)f

^{is a}

positive

function whenever

f C- L2(lt)

^is

positive

^and

[The ^proof

of this result was

inspired by

^a recent paper of L. Monauni

[6].

In

fact, using implicitely

^{Lemma 1}

below,

^he characterizes the

growth

bound _mo for

semigroups

^on Hilbert spaces

by

the boundedness of the

resoi-

vent

along imaginary

^{axes A}

+ iR,

^A &#x3E; For

positive semigroups

^on

L2-spaces

^we obtain this

property

^{from the}

integral representation

^stated

in Lemma 2. For the basic

concepts

^on

one-parameter semigroups

^{we refer}

to

[1]

^or

[2].

LEMMA 2. Let H be a Hilbert space. Then the vector-valued Fourier trans-

form Y, .L2(IL~, H) -~ L2(ll~, H) defined by

^{.F’ H}

.F,

and suitable F E

.L2(lE~, H),

^{is an}

isometry.

PROOF.

Every

^Hilbert

space H

^is

isomorphic

^{to for some measure}

space

(X,

^{Given a} Banach space G and ^an

operator

^{T E}

C(G),

^then

F(.) t-~ T(~(’))

^{defines an}

operator Id ~ T

^on

L2(fl; G) satisfying T~~

In

particular, Id@ T

^{is an isometric}

isomorphism

^{if the same is}

true for T.

Now,

observe that the Hilbert _space

L2(R; H) -

is

canonically isomorphic

^to

L2 (R)). ^Moreover,

^this

isomorphism

^trans-

forms into where Y denotes the scalar-valued Fourier trans- form on

L2(R).

Since Y is

unitary

the assertion follows.

LEMMA 2

([3],

^{3.2 or}

^[4], 3.3 )..Let ^strongly

^continuous

semigroup of positive operators

^{on a} Banach lattice E and denote

by (A, D(A))

its

generator.

Then the resolvent

integral

exists

for every ~,

^{E C}

satisfying

^{Re A} &#x3E;

s(A).

^In

particular,

the resolvent

:=

(A

^-

A)-’,

^{Â E}

satisfies 11 IIR(Re Â) II for

every A c- C with Re  &#x3E;

s(A).

THEOREM. Let be a

strongly

continuous

semigroup of positive

operators

^on

.L2(,u).

^{Then the}

spectral

^bound

s(A) of

^the

generator

A coincides with the

growth

^{bound 000}

of ~T(t)~t~o.

PROOF. We _may assume that 0 which

implies

^that

s(A)

^{c0. For}

every with

Re a &#x3E; s(A),

^and

f

^E

.L2(,u),

^{we define}

continuous, L2(p,)-

valued functions

For Re a &#x3E; 0 it follows from the definition of 000 that is contained in

L2(R,

ⁿ

L2(p)). ^Using

^the

integral representation

of the resol- vent we infer from

[2],

^{th. 2.8 that}

By

^{Lemma 1 we conclude and obtain}

for some constant CaE R

independent

^of

f.

In order to prove the

assertion,

^{we assume to the}

contrary

^that

s(A)

^0.

Then take and and recall the

following

identity:

Since

6~(’)

is continuous and

JIB(a±iA)II II

is dominated

by IIR(Re a) II (use

^Lemma

2)

^{we obtain}

The inverse Fourier transform

applied

^to

G f gives

^{a new function}

We

already

^{stated in}

⁽¹⁾

^that

H f =

for every ^L-t with Re a &#x3E; 0.

260

Now

keep

^{S E Rand}

f E D(A2)

fixed and observe that _{the map}

is

analytic

^on C. On the other hand the function

is

analytic

^on

fa

^{E C: Re a} &#x3E;

s(A)}.

^{This can be seen} from the theorem of Morera since

for every closed ^curve 1’ contained in the

right semiplane

determined

by s(A).

^Here the order of

integration

may be reversed since the ^norm of the

integrand

^{is dominated}

by 2~,-2( ~~R(Re a) ~~ ’ ~~ (a - A)2 f ~~ + (use

(3)

^{and Lemma}

2).

^{From the}

uniqueness

theorem for

analytic functions,

we conclude that

on the

semiplane {a

^{E C:} &#x3E;

s(A)}. ^Applying

^{Lemma 1}

^again

^{we obtain}

for and

f E D(A2) .

For Bex&#x3E;0 we could estimate

by

^{a suitable}

multiple

^of

lifli 2 (see (2)).

^{In order to obtain an}

^analogous

estimate for

we use the

following

^{relation between}

R(Re a)

^and

.R(-

^Re

a) .

^Choose

0 - Lx - s(A)

^{in such a} way that

(1- 2a.R(a) )

^becomes invertible.

Using

^{the resolvent}

equation

in the form

one deduces the

identity

which

yields

the estimate

for every

f

^E

D(A2)

^{and some constant}

^da independent

^of

f.

Putting together (4), (5)

^and

(2)

^{we have}

finally

for every

f E D(A2)

^{and a constant}

da c_ a

^still

independent

^of

f.

Since

D(A2)

is dense in we may extend this ectimate and obtain

for

By

Datko’s theorem

(see [8],

^p.

121)

^this

^implies

0

contradicting

^our

assumption.

Therefore the

spectral

^bound

s(A)

has to be 0.

FINAL REMARK. While

u s(A)

⁼ holds for

positive semigroups

^on

(see [3], 3.3),

^{on and on}

([3], 3.3,

^also

[5])

^{we do not know}

whether the statement still holds for 1 p oo, 2.

REFERENCES

[1]

^A. BELLENI-MORANTE,

Applied Semigroups

and Evolution

Equations,

^Oxford

University

Press, Oxford, ^1979.

[2] ^{E. B.} DAVIES,

One-parameter Semigroups,

^Academic Press, London, ^1980.

[3] ^R. DERNDINGER, Über das

Spektrum positiver Generatoren,

^{Math. Z., 172}

(1980),

pp. 281-293.

[4] ^G. GREINER, ^J. VOIGT, ^M. WOLFF, ^{On the}

spectral

^bound

of

^the generator

of

^semi-

groups

of positive

operators, ^J.

Operator Theory,

⁵ (1981), pp. 245-256.

[5] ^U. GROH, ^F. NEUBRANDER, Stabilität

starkstetiger, positiver Operatorhalbgruppen

auf C*-Algebren,

^{Math. Ann., 256}

(1981),

pp. 509-516.

262

[6] ^{L. A.} MONAUNI, On the abstract

Cauchy problem

^{and the} generation

problem for semigroups of

^bounded

operators,

^Control

Theory

^Centre

Report

^No. 90, War- wick, ^1980.

[7] ^R. NAGEL, ^Zur

Charakterisierung

^stabiler

Operatorhalbgruppen,

Semesterbericht

Funktionalanalysis, Tübingen,

^1981-82.

[8] ^A. PAZY,

Semigroups of

^linear

operators

^and

applications

^to

partial differential equations,

^Univ.

Maryland

Lecture Notes No. 10, ^1974.

[9] ^{A. J.} PRITCHARD, ^J. ZABCZYK,

Stability

^and

stabilizability of infinite

^dimen-

sional systems, ^SIAM Rev., ²³

(1981),

pp. ^25-52.

[10] ^J. ZABCZYK, ^A note on

C0-semigroups,

^Bull. Acad. Polon. Sci., ²³

(1975),

pp. 895-898.

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