Almost periodic and strongly stable semigroups of operators

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 ${\displaystyle \{T^n:n=0,1,...\}}$ on a Banach space X (discrete one-parameter semigroups), one-parameter ${\displaystyle C_0}$-semigroups ${\displaystyle \{T\left(t\right):t\ge 0\}}$ 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 ${\displaystyle C_0}$-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 ${\displaystyle C_0}$-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 ${\displaystyle C_0}$-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.