Semi-groups and collectively compact sets of linear operators

John Deryck De Pree and Harry Scott Klein

Vol. 55 (1974), No. 1, 55–63

DOI: 10.2140/pjm.1974.55.55

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 T(t₁ + t₂) = T(t₁)T(t₂) 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.

Mathematical Subject Classification

Primary: 47D05

Milestones

Received: 4 April 1973

Published: 1 November 1974