On commuting isometries and commuting isometric semigroups

Abstract

This thesis consists of two parts- commuting isometries and commuting isometric semigroups. In the first part, we study the Taylor joint spectrum for a pair of commuting isometries in certain cases. We show that the joint spectrum of two commuting isometries can range from being small (of measure zero or an analytic disc for example) to the full bidisc. En route, we discover a new model pair in the negative defect case and relate it to the modified bi-shift. The rest of the thesis focuses on the study of a pair of commuting C_0-semigroups of isometries. An analogue of the Wold decomposition for an isometry is the Cooper’s decomposition for a C_0 -semigroup of isometries. This decomposition provides a comprehensive structure for a C_0 -semigroup of isometries. We discover a structure for a pair of commuting C_0 -semigroups of isometries in generality as well as under certain additional assumptions like double commutativity or dual double commutativity. Cooper showed that the role of the unilateral shift in the Wold decomposition of an isometry is played by the right-shift-semigroup for a C_0 -semigroup of isometries. The factorizations of the unilateral shift have been explored by Berger, Coburn, and Lebow. We give a complete description of the factorizations of the right-shift-semigroup under the assumption of the multiplicity space being finite dimensional. We employ novel function theoretic methods and classical convex analysis to arrive at the factorization.