The Structure of C(0) Operators.

This work is part of a larger project which consists in an exposition of all the major results obtained for the class of C(,0) operators. An operator T acting on a Hilbert space belongs to the class C(,0) if it is a completely nonunitary contraction ((PARLL)T(PARLL) (LESSTHEQ) 1 and none of the restrictions of T is unitary) and u(T) = 0 for some bounded analytic function u in the unit disc. In this work we restrict ourselves to those parts of the theory of C(,0) operators in which the general model theory of Sz.-Nagy and C. Foias is not needed. In particular, Section 1 of Chapter I may be considered as an introduction to canonical model theory via a very elementary class of operators. In the remaining sections of this chapter we introduce the class C(,0) and study its elementary properties. Chapter II concerns the quasisimilarity classification theorem for operators of class C(,0) (quasisimilarity is an equivalence relation for operators, weaker than similarity). This classification theorem is similar in many ways to the classical theory of Jordan, concerning operators on finite dimensional spaces, but it is blended with elements of the multiplicity theory for normal operators. In Chapter III we give various applications of the structure theorem for operators of class C(,0). Here we study the relations between the various algebras related with a given operator of class C(,0). We also consider reflexivity of C(,0) operators, and their lattices of hyperinvariant subspaces. The Appendix is dedicated to a historical sketch of those parts of the theory of C(,0) operators that are covered in this work. In particular the author's contributions to the subject are singled out.PhDMathematicsUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/158957/1/8224911.pd