The invariant subspace problem is the long-standing question whether every operator on a Hilbert space of dimension greater than one has a non-trivial invariant subspace. Although the problem is unsolved in the Hilbert space case, there are counter-examples for operators acting on certain well-known non-reflexive Banach spaces. These counter-examples are constructed by considering a single orbit and then extending continuously to a hounded linear map on the entire space. Based on this process, we introduce an operator which has properties closely linked with an orbit. We call this operator the orbit operator. In the first part of the thesis, examples and basic properties of the orbit operator are discussed. Next, properties linking invariant subspaces to properties of the orbit operator are presented. Topics include the kernel and range of the orbit operator, compact operators, dilation theory, and Rotas theorem. Finally, we extend results obtained for strict contractions to contractions.

Thesis (Ph. D.)--University of Missouri-Columbia, 2007. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on March 11, 2008) Includes bibliographical references.

Thesis (M.C.S.)--Carleton University, 2001. / Includes bibliographical references (p. 68-72). Also available in electronic format on the Internet.

Thesis--Bonn. Extra t.p. with thesis statement inserted. / Includes bibliographical references (69-72).

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [99]-104).

Thesis--Bonn. Extra t.p. with thesis statement inserted. / Includes bibliographical references (69-72).

Thesis (M. Phil.)--University of Hong Kong, 1993.

Thesis (M.S.)--Georgia Southern University, 2005. / "A thesis submitted to the Graduate Faculty of Georgia Southern University in partial fulfillment of the requirements for the degree Master of Science" ETD. Includes bibliographical references (p. 131-132)

The main study of this thesis is the holomorphic functional calculi for three classes of unbounded operators: sectorial, strip-type and half-plane. The functional calculus for sectorial operators was introduced by McIntosh as an extension of the Riesz-Dunford model for bounded operators. More recently Haase has developed an abstract framework which incorporates analogous constructions for strip-type and half-plane operators. These operators are of interest since they arise naturally as generators of C₀-(semi)groups. The theory of bounded H^∞-calculus for sectorial operators is well established and it has been found to have many applications in operator theory and parabolic evolution equations. We survey these known results, first on Hilbert space and then on general Banach space. Our main goal is to fill the gaps in the parallel theory for strip-type operators. Whilst some of this can be deduced by taking exponentials and applying known results for sectorial operators, in general this is insu_cient to obtain our desired results and so we pursue an independent approach. Starting on Hilbert space, we broaden known characterisations of the bounded H^∞-calculus for strip-type operators by introducing a notion of absolute calculus which is an analogue to the established notion for the sectorial case. Moving to general Banach space, we build on the work of Vörös, broadening his characterisation for strip-type operators in terms of weak integral estimates by introducing a new, but equivalent, notion of the bounded H^∞-calculus, which we call the m-bounded calculus. We also demonstrate that these characterisations fail for half-plane operators and instead present a weaker form of the bounded H-calculus which is more natural for these operators. This allows us to obtain new and simple proofs of well known generation theorems due to Gomilko and Shi-Feng, with extensions to polynomially bounded semigroups. The connection between the bounded H-calculus of semigroup generators and polynomial boundedness of their associated Cayley Transforms is also explored. Finally we present a series of results on sums of operators, in connection with maximal regularity. We also establish stability results for the bounded H^∞-calculus for strip-type operators by showing it is preserved under suitable bounded perturbations, which at time requires further assumptions on the underlying Banach space. This relies heavily on intermediate characterisations of the bounded H^∞-calculus due to Kalton and Weis.

Ph.D. (Mathematics) / Our objective in this thesis is to investigate two fundamental questions concerning Riesz operators de ned on a Banach space. Recall that Riesz operators are generalizations of compact operators in the sense that Riesz operators have the same spectral properties as compact operators. However, they do not possess the same algebraic properties as compact operators. Our rst question that we investigate is: When is a Riesz operator a nite rank operator? This question is motivated from the fact that if a compact operator de ned on a Banach space has closed range, then it is a nite rank operator. Also, Ghahramani proved that a compact homomorphism de ned on a C -algebra is a nite rank operator, see . Martin Mathieu, in his paper, generalized the result of Ghahramani by proving that a weakly compact homomorphism de ned on a C -algebra is a nite rank operator...