On the spectral Theory of Operator Measures

TL;DR

This paper advances the spectral theory of operator measures by solving foundational problems, introducing multiplicity functions, and analyzing principal vectors and spectral types, with implications for dilation and boundedness.

Contribution

It provides a new inner description of $L_2$ spaces for operator measures, introduces the multiplicity function, and characterizes principal vectors and spectral types, addressing longstanding questions.

Findings

Solved Krein's problem on $L_2(\Sigma,H)$ space

Established criteria for spectral measure dilations

Proved density of principal vectors in cyclic subspaces

Abstract

In the first section we provide a solution to the M. G. Krein problem about an inner description of the space $L_2(\Sigma,H).$ In the second section we introduce the multiplicity function for an operator measure. Making use of the description of the space $L_2(\Sigma,H)$ we establish the correctness of the definition and give a criterion for a spectral measure to be a dilation of a given operator measure. In the third section we prove that the set of principal vectors of an operator measure is an everywhere dense $G_\delta.$ This implies, in particular, that there are a lot of principal vectors in any cyclic subspace of a selfadjoint operator. In the 4th section we introduce Hellinger spectral types for an arbitrary operator measure and prove the existence of subspaces, realizing them. In the 5-th section we give an answer to an old question of Paulsen and provide an analytic description of completely bounded operator measures. The results of this section belong to the second author.