Functional Model and the Unitary Invariants of Self-adjoint Operators

Abstract

In this chapter a functional model for an arbitrary spectral measure on Hilbert space is constructed. It allows us to describe the complete system of unitary invariants of a spectral measure. Multiplication operators on direct integrals of Hilbert spaces play an important role in what follows. Sections 1–2 are devoted to the construction of the direct integral and to the study of multiplication operators on it. In Sections 3–4 we show how an arbitrary spectral measure can be reduced to its model. In Section 5 we consider the functional model and the unitary invariants of a finite family of commuting self-adjoint operators (and of one operator in particular). Because of the spectral representations of Chapter 6 the results are direct consequences of the corresponding results for spectral measures. Our exposition provides the immediate generalization to the case of normal operators. Section 6 deals with the decomposition of a spectral measure into the absolutely continuous and singular parts with respect to a given scalar measure.