Date of Award

Spring 2005

Project Type

Dissertation

Program or Major

Mathematics

Degree Name

Doctor of Philosophy

First Advisor

Donald Hadwin

Abstract

In the first part of the thesis we obtain some new results in Hadwin's general version of reflexivity and apply them in the classical cases. We prove that the image of any C*-algebra under any bounded unital homomorphism into B(W) is approximately reflexive, where W is a Banach space. We also introduce two new versions of reflexivity, approximate algebraic reflexivity and asymptotic reflexivity, and study their properties.

In the second part of the thesis we construct a general setting in which functions of bounded mean oscillation (BMO) and vanishing mean oscillation (VMO) can be studied. In this setting we prove a version of the John-Nirenberg theorem and a version of Sarason's characterization of VMO as the closure of the uniformly continuous functions in BMO. We also prove that VMO is never complemented in BMO.

In the third part of the thesis we study the stable invariant subspaces of Hilbert-space operators. We prove that if T is an operator on an infinite dimensional Hilbert space whose spectrum and essential spectrum are both connected and whose Fredholm index is only 0 or 1, then the only nontrivial norm-stable invariant subspaces of T are the finite dimensional ones. We also characterize norm-stable invariant subspaces of any weighted unilateral shift operator.

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