Almost periodic and strongly stable semigroups of operators

Vũ Phóng

Banach Center Publications (1997)

  • Volume: 38, Issue: 1, page 401-426
  • ISSN: 0137-6934

Abstract

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This paper is chiefly a survey of results obtained in recent years on the asymptotic behaviour of semigroups of bounded linear operators on a Banach space. From our general point of view, discrete families of operators T n : n = 0 , 1 , . . . on a Banach space X (discrete one-parameter semigroups), one-parameter C 0 -semigroups T ( t ) : t ≥ 0 on X (strongly continuous one-parameter semigroups), are particular cases of representations of topological abelian semigroups. Namely, given a topological abelian semigroup S, a family of bounded linear operators T(s): s ∈ S is called a representation of S in B(X) if: (i) T(s+t) = T(s)T(t); (ii) For every x ∈ X, s ↦ T(s)x is a continuous mapping from S to X. The central result which will be discussed in this article is a spectral criterion for almost periodicity of semigroups, obtained by Lyubich and the author [40] for uniformly continuous representations of arbitrary topological abelian semigroups (thus including the case of single bounded operators and several commuting bounded operators), and for C 0 -semigroups [41], and by Batty and the author [9] for arbitrary strongly continuous representations of suitable locally compact abelian semigroups. An immediate consequence of this result is a Stability Theorem, obtained, for single operators and C 0 -semigroups, also by Arendt and Batty [1] independently. The proof in [1] uses a Tauberian theorem for the Laplace-Stieltjes transforms and transfinite induction. Methods of this type can also be used to prove the almost periodicity result for C 0 -semigroups [8], but seem not suitable for commuting semigroups, and will not be discussed in this article. We also refer the reader to a recent survey article of Batty [6], where some developments are described which are not included here.

How to cite

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Phóng, Vũ. "Almost periodic and strongly stable semigroups of operators." Banach Center Publications 38.1 (1997): 401-426. <http://eudml.org/doc/208644>.

@article{Phóng1997,
abstract = {This paper is chiefly a survey of results obtained in recent years on the asymptotic behaviour of semigroups of bounded linear operators on a Banach space. From our general point of view, discrete families of operators $\{T^\{n\}: n = 0,1,... \}$ on a Banach space X (discrete one-parameter semigroups), one-parameter $C_0$-semigroups $\{T(t): t ≥ 0\}$ on X (strongly continuous one-parameter semigroups), are particular cases of representations of topological abelian semigroups. Namely, given a topological abelian semigroup S, a family of bounded linear operators T(s): s ∈ S is called a representation of S in B(X) if: (i) T(s+t) = T(s)T(t); (ii) For every x ∈ X, s ↦ T(s)x is a continuous mapping from S to X. The central result which will be discussed in this article is a spectral criterion for almost periodicity of semigroups, obtained by Lyubich and the author [40] for uniformly continuous representations of arbitrary topological abelian semigroups (thus including the case of single bounded operators and several commuting bounded operators), and for $C_0$-semigroups [41], and by Batty and the author [9] for arbitrary strongly continuous representations of suitable locally compact abelian semigroups. An immediate consequence of this result is a Stability Theorem, obtained, for single operators and $C_0$-semigroups, also by Arendt and Batty [1] independently. The proof in [1] uses a Tauberian theorem for the Laplace-Stieltjes transforms and transfinite induction. Methods of this type can also be used to prove the almost periodicity result for $C_0$-semigroups [8], but seem not suitable for commuting semigroups, and will not be discussed in this article. We also refer the reader to a recent survey article of Batty [6], where some developments are described which are not included here.},
author = {Phóng, Vũ},
journal = {Banach Center Publications},
keywords = {Abelian semigroup; representation; -semigroup; stability; asymptotics; almost periodicity},
language = {eng},
number = {1},
pages = {401-426},
title = {Almost periodic and strongly stable semigroups of operators},
url = {http://eudml.org/doc/208644},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Phóng, Vũ
TI - Almost periodic and strongly stable semigroups of operators
JO - Banach Center Publications
PY - 1997
VL - 38
IS - 1
SP - 401
EP - 426
AB - This paper is chiefly a survey of results obtained in recent years on the asymptotic behaviour of semigroups of bounded linear operators on a Banach space. From our general point of view, discrete families of operators ${T^{n}: n = 0,1,... }$ on a Banach space X (discrete one-parameter semigroups), one-parameter $C_0$-semigroups ${T(t): t ≥ 0}$ on X (strongly continuous one-parameter semigroups), are particular cases of representations of topological abelian semigroups. Namely, given a topological abelian semigroup S, a family of bounded linear operators T(s): s ∈ S is called a representation of S in B(X) if: (i) T(s+t) = T(s)T(t); (ii) For every x ∈ X, s ↦ T(s)x is a continuous mapping from S to X. The central result which will be discussed in this article is a spectral criterion for almost periodicity of semigroups, obtained by Lyubich and the author [40] for uniformly continuous representations of arbitrary topological abelian semigroups (thus including the case of single bounded operators and several commuting bounded operators), and for $C_0$-semigroups [41], and by Batty and the author [9] for arbitrary strongly continuous representations of suitable locally compact abelian semigroups. An immediate consequence of this result is a Stability Theorem, obtained, for single operators and $C_0$-semigroups, also by Arendt and Batty [1] independently. The proof in [1] uses a Tauberian theorem for the Laplace-Stieltjes transforms and transfinite induction. Methods of this type can also be used to prove the almost periodicity result for $C_0$-semigroups [8], but seem not suitable for commuting semigroups, and will not be discussed in this article. We also refer the reader to a recent survey article of Batty [6], where some developments are described which are not included here.
LA - eng
KW - Abelian semigroup; representation; -semigroup; stability; asymptotics; almost periodicity
UR - http://eudml.org/doc/208644
ER -

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