Triples of operator-valued functions related to the unit circle

Abstract

In this paper various triples of operator-valued functions acting in a Hubert space are characterized, and the members are shown to be connected by a one-to-one-to-one correspondence. The elements of the triples are operator measures, generalized resolvents, and positive definite sequences which are related to the unit circle. The relationships between operator measures and positive definite sequences were first obtained by M. A. Naimark and B.Sz.-Nagy in their dilation and moment theorems. The main contribution of this paper is a characterization of the interrelated resolvent classes. By exploiting the correspondence between the various classes, a unified development of the theory is obtained.