Vol. 55, No. 1, 1974

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Semi-groups and collectively compact sets of linear operators

John Deryck De Pree and Harry Scott Klein

Vol. 55 (1974), No. 1, 55–63

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
Received: 4 April 1973
Published: 1 November 1974
John Deryck De Pree
Harry Scott Klein