ON THE COLLECTIVE COMPACTNESS OF STRONGLY CONTINUOUS SEMIGROUPS AND COSINE FUNCTIONS OF OPERATORS

Abstract

Let $X$ be a complex Banach spcce, and denote by $T$ a strongly continuous semigroup of linear operators defined on $X$ and by $C$ a cosine function of operators with associated sine function $S$ defined on $X$. In this note we characterize in terms of spectral properties of the infinitesimal generator those semigroups $T$ and cosine functions $C$ such that $\{T(t) - I : t \geq 0 \}$, $\{C(t) - I : t \in {\Bbb R}\}\;$ and $\{S(t) : t \in {\Bbb R}\}$ are collectively compact sets of bounded linear operators.