Vol. 61, No. 2, 1975

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Some generalizations of Schauder’s theorem in locally convex spaces

Volker Wrobel

Vol. 61 (1975), No. 2, 579–586

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
Received: 22 July 1975
Revised: 13 November 1975
Published: 1 December 1975
Volker Wrobel