Spectral measures. II: Characterization of scalar operators (Q1059835)

This is the second of a series of papers which have appeared in this journal on the theory of spectral measures having values in general topological algebras [for part I see ibid. 13, 205-227 (1982; Zbl 0515.47012)]. The main tool in this paper is the following version of the Riesz representation theorem: a continuous linear map \(\Lambda: C_ 0(X)\to Y,\) where X is a locally compact Hausdorff space and Y an arbitrary topological vector space, can be represented as an integral with respect to a countably additive regular Borel measure with values in Y. Using this result the author discusses various topics: products of spectral measures, the generation of spectral measures and functions of several commuting operators which are of scalar-type in the sense of Dunford. He also gives a Radon-Nikodym theorem for spectral measures.