Semi-groups and collectively compact sets of linear operators

Abstract

A set of linear operators from one Banach space to another is collectively compact if and only if the union of the images of the unit ball has compact closure. Semi-groups S = {T(t): t ^ 0} of bounded linear operators on a complex Banach space into itself and in which every operator T(t), t > 0 is compact are considered. Since Tit, + t 2) = T(tJT(tJ for each operator in the semi-group, it would be expected that the theory of collectively compact sets of linear operators could be profitably applied to semi-groups. 1* Introduction* Let X be a complex Banach space with unit ball Xλ and let [X, X] denote the space of all bounded linear operators on X equipped with the uniform operator topology. The semi-group definitions and terminology used are those of Hille and Phillips [6]. Let S be a semi-group of vector-valued functions T: [0, oo)—>[X, X]. It is assumed that T(t) is strongly continuous for t ^ 0. If lim^ || T(t)x - T(to)x || = 0 for each t0 ^ 0, xeX and if there is a constant M such that the || T(t) || ^ M for each t ^ 0, then S — {T(t): t ^> 0} is called an equicontinuou s semi-group of class Co. The infinitesimal generator A of the semi-group S is defined by Ax = lim— [T(s)x - x] s->o S whenever the limit exists. The domain D(A) of A is a dense subset of X consisting of just those elements x for which this limit exists. A is a closed linear operator having resolvents R(X) which, for each complex number λ with the real part of λ greater than zero, are given by the absolutely summable Riemann-Stieltjes integral