Szegö-type decompositions for isometries

Abstract

The notion of Szegö-type properties of positive Borel measures is well known and widely exploited. In this paper, we consider a class of orthogonal decompositions of isometries on Hilbert spaces which correspond to Szegö-type properties of their elementary measures. Our decompositions are closely connected with some special families of invariant subspaces. It is shown that this connection holds for the decomposition constructed in the paper. We illustrate our results with several examples. We also give a short proof of Mlak’s theorem on the elementary measures of completely nonunitary contractions.