Quasi-compact Operators in Topological Linear Spaces

The classical theorems of Riesz [l] on compact operators have been extended by Leray [2] and Williamson [3] to the context of topological linear spaces. Ringrose [4] has shown that if an operator on such a space is compact, the square of its adjoint is also compact, where the topology on the dual space is that of uniform convergence on bounded sets. Thus if an operator is continuous and some power is compact, its adjoint shares the same property. We shall call such operators quasi-compact; in this note we prove the Riesz theorems for quasi-compact operators in topological linear spaces. This has already been done for Banach spaces by Zaanen [5, Chapter 11 ]. "Space" will mean a Hausdorff topological linear space. Most of our definitions are as in Bourbaki [6], [7]. A set E is circled if for every complex number e such that |e| ^1, eEEE. The circled neighborhoods of 0 form a base of neighborhoods at 0. Hereafter, "neighborhood" will mean "circled neighborhood of 0"; P will be a continuous operator on a space X such that Tr is compact; W will denote an open neighborhood such that the closure (T'W)~~ is compact, and U = ~k — T, where X is a nonzero scalar.