Let E1, E2, E8, E4 be four
locally convex Hausdorff spaces (l.c.s.); denote by ℒb(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) : ℒb(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.
