CHAPTER 2 - Self-adjoint operators in spaces of functions of an infinite number of variables

Abstract

This chapter describes self-adjoint operators in spaces of functions of an infinite number of variables. It deals with only one of the questions concerning the theory of operators acting in spaces of an infinite number of variables. The spectral representation of a family of commuting normal operators which are connected by correlations of various kinds is also analyzed. The chapter describes another way in which classical Stone theorem on spectral representation can be considered. There exists a family of commuting unitary operators (Ax)x∈R1 which continuously depend on x and satisfy the operational functional equation Ax+y = AxAy (x, y ∈ R1). Then Ax = ∫R1eiλx dE(λ) (x ∈ R1), where E is some spectral measure. It is shown in this chapter how to obtain similar representations in case the above-mentioned functional equation has a rather general structure, for instance, when it is a differential equation, etc.