St. Petersburg Mathematical Journal

Abstract:

In §2 the spaces $L^2(\Sigma ,H)$ are described; this is a solution of a problem posed by M. G. Kreĭn.

In §3 unitary dilations are used to illustrate the techniques of operator measures. In particular, a simple proof of the Naĭmark dilation theorem is presented, together with an explicit construction of a resolution of the identity. In §4, the multiplicity function $N_{\Sigma }$ is introduced for an arbitrary (nonorthogonal) operator measure in $H$. The description of $L^2(\Sigma ,H)$ is employed to show that this notion is well defined. As a supplement to the Naĭmark dilation theorem, a criterion is found for an orthogonal measure $E$ to be unitarily equivalent to the minimal (orthogonal) dilation of the measure $\Sigma$.

In §5 it is proved that the set $\Omega _{\Sigma }$ of all principal vectors of an arbitrary operator measure $\Sigma$ in $H$ is massive, i.e., it is a dense $G_{\delta }$-set in $H$. In particular, it is shown that the set of principal vectors of a selfadjoint operator is massive in any cyclic subspace.

In §6, the Hellinger types are introduced for an arbitrary operator measure; it is proved that subspaces realizing these types exist and form a massive set.

In §7, a model of a symmetric operator in the space $L^2(\Sigma ,H)$ is studied.