WOLD-TYPE DECOMPOSITION AND THE STRUCTURE FOR TUPLES OF CONTRACTIONS AND ISOMETRIES

Abstract

A fundamental challenge in the theory of operators, operator algebras, and function theory is the classification of tuples of contractions and isometries acting on Hilbert spaces. Many significant theorems in operator theory focus on representing abstract operators through concrete operators in well-understood spaces. For example, the spectral theorem represents normal operators as multiplication operators in Lebesgue spaces. Additionally, several important representation theorems exist for non-normal operators, where such operators are expressed as multiplication operators on specific Hilbert spaces of analytic functions. Sz.-Nagy and Foia¸s’ theorem extends this approach by modelling certain contractions as compression of unilateral shifts on model spaces. This line of research has been remarkably successful, leading to the study of model spaces, from scalar-valued to vector-valued, and broadening the scope of operator theory. This thesis focuses on the orthogonal decomposition for a large class of operator tuples on Hilbert spaces. It begins with introductory material and a literature review on the decomposition of various non-normal operators. Additionally, well-established results concerning the canonical decomposition of certain non-normal operators, such as contractions, isometries, and power partial isometries, are reviewed.