Unitary Representations and Regularity for Self-adjoint Operators

Abstract

In this chapter we specialize some of the considerations of Chapter 5 to the case of unitary C ₀-Groups in a Hubert space H. The theory of unitary representations $W(x) = {e^{iA - x}}$ of ℝⁿ is a very well understood classical subject and will not be presented here. However we mention that a n-dimensional version of Stone’s theorem states that there is a unique spectral measure E on ℝⁿ such that $W(x) = \int_{{^n}} {{e^{ix \cdot y}}E(dy)}$ and this allows one to extend the functional calculus which we already have for functions in $C_{pol}^\infty ({^n})$ to all Borel functions $\varphi :{^n} \to$. The natural definition of φ(A) for a Borel function φ is $\varphi :(A) = \int_{{^n}} {\varphi (y)E(dy)}$, and one may check that for $\varphi \in C_{pol}^\infty ({^n})$ the two definitions lead to the same operator φ(A).