Spectral Representations and Spectral Functions of Symmetric Operators

Abstract

The spectral representation of a symmetric nonmaximal operator transforms it into a multiplication operator by a complex variable acting in a linear space of pairs of vectorvalued functions holomorphic on the upper and lower half- planes. Each generalized spectral function of the same symmetric operator leads to its realization in form of a multiplication operator by a real variable in some Hilbert space. We study the connection between such realizations of the given operator and its spectral representation.