Boolean Algebras of Projections on Banach Spaces

[摘要] The object of this thesis is the study of the Boolean algebras of projections on Banach spaces and their properties. Chapter one is devoted to the study of the integration theory needed in the other chapters. The main characteristic features of the theory of normed KOthe spaces are introduced first. Next, we investigate the conditions on p which ensure that every function 0 6 L1(mu) can be factorized as &phis; = f.g, with f ∈ Lp and g ∈ Lp'. In chapter two, which is the main part of the thesis, we introduce the concept of Boolean algebras of projections. We prove that a Boolean algebra of projections is a-complete if and only if it is the range of a countably additive regular spectral measure. We show that a bounded Boolean algebra of projections on a weakly complete Banach space X can be embedded in a alpha-complete Boolean algebra of projections on X. Next, the main structure theorem for a cyclic Banach space X is proved. It is shown that the weakly closed algebra generated by a a-complete Boolean algebra of projections on a Banach space X is reflexive. A representation theorem for a complete Boolean algebra of projections is included. In chapter three, we consider the necessary and sufficient conditions for the restrictions of normal operators on Hilbert space to be normal operators, and the restrictions of scalar-type spectral operators on Banach spaces to be of scalar-type. The results of chapter one are due to T.A. Gillespie [13] and [14]. The majority of the results in chapter two are due to W.G. Bade [1] and [9], but we show how Gillespie's results can be used to obtain stronger results and give more satisfactory proofs of these results. The results of Chapter three are due to Wermer [18] and Dawson [5].

[发布机构] University:University of Glasgow

[关键词] Mathematics