On extensions of positive definite operator-valued functions (Q1823415)

Generalizing results of Krein, Livshits, Berezanskij et al., the authors show the existence of positive definite extensions \(\tilde T\) on \(G\times reals\) of T: \(G\times (-a,a)\to \{bounded\) linear operators in a Hilbert space \(K\}\) for any abelian topological group G. This is done with commuting selfadjoint resp. unitary extensions of a closed symmetric A (infinitesimal real translation) resp. unitary representation U, both defined via T. Conditions on \(\tilde T\) for uniqueness are given; especially if T(0,\(\cdot)\) has only one p.d. extension, so has T. For locally compact G the positive definite T are described with operator- valued measures on the dual \(\hat G;\) if K is finite-dimensional, this can be done even with a ``channel function'' as introduced by Levin and Ovcharenko for \(G={\mathbb{R}}^ n\). Finally the authors give an explicit one-to-one correspondence between all positive definite extensions \(\tilde T\) and the generalized resolvents of A which commute with U; for dim K\(=1\), this gives a formula for all \(\tilde T\) with five arbitrary functions holomorphic on Re z\(>0\). Several examples complement these results.