Vol. 61, No. 2, 1975

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ISSN: 0030-8730
Some generalizations of Schauder’s theorem in locally convex spaces

Volker Wrobel

Vol. 61 (1975), No. 2, 579–586
Abstract

Let E1, E2, E8, E4 be four locally convex Hausdorff spaces (l.c.s.); denote by b(Ei,Ek) the set of all continuous linear operators from Ei into Ek with the topology of uniform convergence on bounded subsets of Ei. Given two linear operators f ∈ℒ(E1,E2) and g ∈ℒ(E8,E4), consider the generalized adjoint operator Hom(f,g) : b(E2,E3) →ℒb(E1,E4) defined by u Hom(f,g)u = g u f. This paper deals with transformation properties of Hom(f,g) and their interactions with those of f and g. This purpose may be illustrated by a result due to K. Vala which generalizes Schauder’s well-known theorem concerning precompact operators and their adjoints on normed spaces: Let all spaces under consideration be normed, let f and g both be nonzero. Then Hom(f,g) is a precompact operator if and only if f and g are precompact operators. In the present paper bounded and precompact operators on l.c.s. are investigated.

Mathematical Subject Classification 2000
Primary: 47A05
Secondary: 47B05
Milestones
Received: 22 July 1975
Revised: 13 November 1975
Published: 1 December 1975
Authors
Volker Wrobel