On Decomposition of Operators in Hilbert Spaces

Abstract

In this project, we investigate the direct sum decomposition of some classes of operators in Hilbert spaces with the aim of de ning properties of the direct summands of these operators. We show that an arbitrary operator T decomposes into a normal and a completely nonnormal parts. The properties for which an operator T has nontrivial normal and direct summands are given. In addition, we study this decomposition of operators in some equivalence classes (similar, unitarily equivalent, quasisimilar and almost-similar) of operators. We also investigate the properties of the direct decomposition of a contraction into a unitary and a completely nonunitary parts. We show that an arbitrary operator T decomposes this way upon dividing the operator by its norm (re-normalization).