Spectral theory of operator measures
by
M. Malamud
Donetsk National University, Ukraine
Coauthors: S. Malamud

Let H be a separable Hilbert space, B(H) the space of bounded linear operators on H and let \Sigma be a nondecreasing operator function on R with values in B(H).

With each function \Sigma it is naturally connected the semi-Hilbert space [L\tilde]²(\Sigma, H) and the corresponding Hilbert space L²(\Sigma, H).

Let C₀₀(H) be the set of all strongly continuous compactly supported vector-functions f ranging in finite-dimensional subspaces of H (the subspace depends on f). Then [L\tilde]²(\Sigma, H) is a completion of C₀₀(H) eqquiped with the semi-definite inner product (f, g)_{L²(\Sigma, H)} = \int_R(d\Sigma(t)f(t), g(t))_H. (The intergal is understood as the limit of Riemannian sums). The space L²(\Sigma, H) is defined as the quotient space: L²(\Sigma, H)=[L\tilde]²(\Sigma, H)/ker||·||.

We obtain an inner description of the space L²(\Sigma, H). This problem was posed by M.G. Krein and (in the special case dimH < \infty) solved by I.S. Kac. Further, we construct a theory of Hellinger spectral types for a nonorthogonal operator measure. We establish the existence of subspaces realizing Hellinger spectral types and in particular the existence of vectors of maximal type.

Some facts are new even for orthogonal measures and for a finite-dimensional space H. We show how the spectral Hellinger types of an operator A=A^* can be found via a cyclic subspace L in Cyc(A). It turnes out that the set of the vectors of maximal type, lying in L is an everywhere dense set of type G_\delta and of second category.

Date received: June 20, 2002