Collectively compact sets of linear operators

Philip Marshall Anselone and Theodore Windle Palmer

Vol. 25 (1968), No. 3, 417–422

DOI: 10.2140/pjm.1968.25.417

Abstract

A set of linear operators from one normed linear space to another is collectively compact if and only if the union of the images of the unit ball has compact closure. This paper concerns general properties of such sets. Several useful criteria for sets of linear operators to be collectively compact are given. In particular, every compact set of compact linear operators is collectively compact. As a partial converse, every collectively compact set of self adjoint or normal operators on a Hilbert space is totally bounded.

Mathematical Subject Classification

Primary: 47.45

Milestones

Received: 27 February 1967

Published: 1 June 1968