Vol. 23, No. 1, 1967

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Kernel representations of operators and their adjoints

F. Dennis Sentilles

Vol. 23 (1967), No. 1, 153–162

If S is a locally compact and Hausdorff space and A is a continuous linear operator from C0(S) into the space C(T) with the supremum norm topology then the Riesz Representation Theorem yields the formula [Af](αj) = sf(y)λ(x,dy), where for each x (x,) is a complex-valued regular Borel measure on S. More generally a study is made of kernel functions λ such that sf(y)λ(,dy) C(T) for f of compact support on S. It is shown that λ(,E) is measurable for each Borel set E and that μ(E) = Tλ(x,E)ν(dαj) is a regular measure on S yielding the adjoint formula Aν = μ. Necessary and sufficient conditions are given on λ so that A∗∗(C(S)) C(T) and that A∗∗ be continuous from C(S)β to C(T)β when S is paracompact. Furthermore, kernel representations of β-continuous operators are studied with applications to semi-groups of operators in C0(S) and C(S)β when S is locally compact.

Mathematical Subject Classification
Primary: 47.25
Secondary: 54.00
Received: 8 August 1966
Published: 1 October 1967
F. Dennis Sentilles