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 ? 0}, {C(t) - I : t ? ?} and {S(t) : t ? ?} are collectively compact sets of bounded linear operators.