Spectral Theory of Operator Measures in Hilbert Space

In §2 the spaces L2(Σ,H) are described; this is a solution of a problem posed by M. G. Krĕın. In §3 unitary dilations are used to illustrate the techniques of operator measures. In particular, a simple proof of the Năımark dilation theorem is presented, together with an explicit construction of a resolution of the identity. In §4, the multiplicity function NΣ is introduced for an arbitrary (nonorthogonal) operator measure in H. The description of L2(Σ, H) is employed to show that this notion is well defined. As a supplement to the Năımark dilation theorem, a criterion is found for an orthogonal measure E to be unitarily equivalent to the minimal (orthogonal) dilation of the measure Σ. In §5 it is proved that the set ΩΣ of all principal vectors of an arbitrary operator measure Σ in H is massive, i.e., it is a dense Gδ-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 L2(Σ, H) is studied.