Linear Algebraic Properties of C_{0}-Operators

Abstract

The theory of Jordan models for contractions is due to B. Sz.-Nagy - C. Foias, B. Moore - E.A. Nordgren, and H. Bercovici - D. Voiculescu. J.A. Ball introduced the class of C0-operators relative to a multiply connected domain. A. Zucchi provided a classification of C0-operators relative to ­a multiply connected domain. Since no analogue of the characteristic function of a contraction is available in that context, that study does not yield some of the results available for the unit disk. In this thesis we use a substitute for the characteristic function, suggested by an analogue of Beurling's theorem provided by M.A. Abrahamse and R.G. Douglas. This allows us to prove a relationship between the Jordan models of a C0-operator relative to ­a multiply connected domain, of its restriction to an invariant subspace, and of its compression to the orthocomplement of that subspace. This thesis is organized as follows. In Chapter 2, by defining a quasi-inner function, we provide a generalized Beurling's Theorem. In Chapter 3, we primarily deal with C0-operators relative to ­a multiply connected domain. Finally, in Chapter 4, we study the modular lattice for C0-Operators relative to the open unit disc.