Vol. 25, No. 3, 1968

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Collectively compact sets of linear operators

Philip Marshall Anselone and Theodore Windle Palmer

Vol. 25 (1968), No. 3, 417–422

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
Received: 27 February 1967
Published: 1 June 1968
Philip Marshall Anselone
Theodore Windle Palmer