On the Structure of (Bcp)-Operators and Related Algebras.
Gregory Frank Robel · Deep Blue (University of Michigan) · 1982
The first chapter of this dissertation is concerned with the structure of operators belonging to the class (BCP) of all completely nonunitary contractions T on (separable, infinite-dimensional, complex) Hilbert space H such that (sigma)(,e) (T)(INTERSECT)D is dominating for H('(INFIN)) (D), where D is the open unit disc in (//C) and (sigma)(,e) is the essential spectrum. One contribution made here is that we establish the "right" definition of this class of operators. Previously, a more restrictive definition had been in use. We first obtain results about the predual Q of the ultraweakly closed algebra A generated by an operator of class (BCP). The main such result is that for any family {{L(,ij)} : i,j (GREATERTHEQ) 1} in Q, there exists a pair of orthogonal sequences {x(,i)} and {y(,j)} in H such that {x(,i) (CRTIMES) y(,j)} = {L(,ij)}. We use this result to obtain a theorem on the realization of various operators as compressions of a (BCP)-operator to a semi-invariant subspace. We obtain some density theorems which amplify these results. We show that any (BCP)-operator can be represented, up to unitary equivalence, as a two-way infinite, lower triangular operator matrix in which the diagonal entries are all of class (BCP). We conclude with some observations pertaining to a specific example. In the second chapter, we extend these methods and results to certain analogous algebras with several generators, and we prove that these algebras are reflexive. Finally, we indicate the applicability of these results to certain weighted bilateral shifts.