Integral representations of weakly compact operators

A. Katsaras and Dar-Biau Liu

Vol. 56 (1975), No. 2, 547–556

DOI: 10.2140/pjm.1975.56.547

Abstract

Let X be a completely regular space and E,F locally convex spaces. Denote by C_rc = C_rc(X,E) the space of all continuous functions f from X into E for which f(X) is relatively compact. Uniformly continuous weakly compact operators from C_rc into F are repreesented by integrals with respect to ℒ(E,F) valued measures on the algebra generated by the zero sets. Necessary and sufficient conditions for an operator to be continuous, with respect to certain topologies, are obtained. A sufficient condition for extending a measure to all Baire sets is given.

Mathematical Subject Classification 2000

Primary: 47B37

Milestones

Received: 20 December 1973

Revised: 10 April 1974

Published: 1 February 1975