Invariant subspaces of compact operators on topological vector spaces

Arthur D. Grainger

Vol. 56 (1975), No. 2, 477–493

DOI: 10.2140/pjm.1975.56.477

Abstract

Let (H,τ) bc a topological vector space and let T be a compact linear operator mapping H into H (i.e., T[V ] is contained in a τ− compact set for some τ− neighborhood V of the zero vector in H). Sufficient conditions are given for (H,τ) so that T has a non-triviaI, closed invariant linear subspace. In particular, it is shown that any complete, metrizable topological vector space with a Schauder basis satisfies the conditions stated in this paper. The proofs and conditions are stated within the framework of nonstandard analysis.

Mathematical Subject Classification 2000

Primary: 47A15

Secondary: 47B05, 02H25

Milestones

Received: 16 November 1973

Revised: 12 June 1974

Published: 1 February 1975