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Proquest Dissertations University of Alberta INVARIANT SUBSPACES OF POSITIVE OPERATORS AND MARTINGALES IN BANACH LATTICES by Hailegebriel Gessesse A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Department of Mathematical and Statistical Sciences Edmonton, Alberta Spring 2009 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your We Votre reference ISBN: 978-0-494-55351-0 Our file Notre reference ISBN: 978-0-494-55351-0 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non­ support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformement a la loi canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. •+• Canada University of Alberta Library Release Form Name of Author: Hailegebriel Gessesse Title of Thesis: Invariant Subspaces of Positive Operators and Martingales in Banach Lattices Degree: Doctor of Philosophy Year This Degree Granted: 2009 Permission is hereby granted to the University of Alberta Library to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly, or scientific research purposes only. The author reserves all other publication and other rights in association with the copyright in the thesis, and except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise re­ produced in any material form whatever without the author's prior written permission. Hailegebriel E. Gessesse Date: University of Alberta Faculty of Graduate Studies and Research The undersigned certify that they have read and recommend to the Faculty of Graduate Studies and Research for acceptance, a thesis entitled Invariant Subspaces of Positive Operators and Martingales in Banach Lattices submitted by Hailegebriel Gessesse in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Dr. Vladimir Troitsky (Supervisor) Dr. Alexander Litvak Dr. Volker Runde Dr. Nicole Tomczak-Jaegermann Dr. Valentina Galvani Dr. Roman Drnovsek (External Reader) Date: To my Parents ABSTRACT This thesis is organized into five Chapters. In Chapter 1, we give an in­ troductory part of this manuscript. In Chapter 2, we present preliminaries of the thesis which include basic definitions and some fundamental concepts and facts in functional analysis which are useful for our discussion in later chapters. Chapter 3 deals with the method of minimal vectors and its applications in constructing invariant subspaces of operators on a Banach lattice. For a positive operator T on an ordered Banach space X, one defines (T] = {S ^ 0 : ST < TS} and [T) = {S ^ 0 : ST ^ TS}. We produce invariant subspaces for (T] under some assumptions on T. In Chapter 4, we extend the method of minimal vectors from Banach lattices to ordered Banach spaces satisfying certain conditions and use the extension to construct invariant subspaces of operators defined on ordered Banach spaces with generating cones. Chapter 5 deals with martingales in Banach lattices. A sequence of contrac­ tive positive projections (En) on a Banach lattice X is said to be a filtration if EnEm = EnAm. A sequence (xn) in X is a martingale if Enxm = xn whenever n < m. We denote by M(X, (£«)) the Banach space of all norm uniformly bounded martingales. We characterize a Banach lattice for which M(X, (En)) is a Banach lattice. We produce a counterexample that M(X, (En)) is not necessarily a Banach lattice. We also prove that if (En) is dense filtration on a KB-space X, then X is embedded in M(X, (En)) as a projection band. The results in Chapters 3 and 4 have been published (see [29] and [30]). ACKNOWLEDGEMENT I would like to express my heart felt gratitude to my supervisor, Dr. Vladimir G. Troitsky, for introducing me the problems on which I worked to prepare this thesis and for his numerous suggestions and improvements. I am also deeply grateful for his steadfast encouragement, constructive feedback, and mathematical insights. Our many hours of discussions have given me an invaluable experience that I carry on in my future career. I also thank him for his financial support over these many years. I would also like to thank my professors at the University of Alberta, Dr. Alexander Litvak, Dr. Volker Runde, and Dr. Nicole Tomczak- Jaegermann for all their help during my stay at the Department of Math­ ematical & Statistical Sciences, University of Alberta. Many thanks to my friends and colleagues who were behind me in writing this thesis. I thank my parents for all their encouragement and prayers. Finally, I would like to acknowledge the University of Alberta and all the fund sources for their financial and other supports. Table of Contents 1 Introduction 1 2 Preliminaries 8 2.1 Banach spaces 8 2.2 Sobolev spaces 10 2.3 Banach lattices 11 2.4 Operators 15 2.5 C*-algebras 19 2.6 Invariant Subspace Problem 21 2.7 Lebesgue-Bochner spaces 25 3 Invariant subspaces of operators on a Banach lattice 27 3.1 Minimal vectors in Banach lattices 28 3.2 Application of the method of minimal vectors 35 3.3 Invariant subspaces of compact-friendly operators 39 4 Invariant subspaces of operators on ordered Banach spaces 44 4.1 Ordered Banach spaces 44 4.2 Positive quasinilpotent operators on Krein spaces 51 4.3 Applications to C(K) and Ck(fl) spaces and to C*-algebras . 53 4.4 Applications to Sobolev spaces Wk'p(Q) 54 4.5 A special case: WliP(fi) 56 4.6 Minimal vectors in spaces with a generating cone 59 4.7 Applications of minimal vector technique 64 4.7.1 Non-unital uniform algebras 64 4.7.2 Sobolev spaces 64 5 Banach lattice martingale spaces 67 5.1 Martingales in Banach lattices 67 5.2 Examples of martingales in Banach lattices 69 5.3 When is M(X, (£„)) a Banach lattice? 70 5.4 Counterexamples for non-Banach lattice martingale spaces . 75 5.5 How does X sit in M(X,(En))? 78 Bibliography 82 Chapter 1 Introduction The Invariant Subspace Problem is one of the most famous and still unsettled problems in functional analysis and modern mathematics. It is stated as a question: When does a bounded operator on a Banach space have a non-trivial closed invariant subspace? The problem was motivated by the Jordan decom­ position of matrices and by the desire to understand the structure and the geometry of an arbitrary operator. P. Enflo [26] found an example of a bounded operator on a separable Ba­ nach space without non-trivial closed invariant subspaces. C. J. Read [44] presented an example of a bounded operator on l\ without non-trivial closed invariant subspaces. Thus, these examples established that the invariant sub- space problem in its general form has a negative answer. However, for various important classes of Banach spaces and operators the Invariant Subspace Prob­ lem remains open. For instance, it is not known if every bounded operator on a separable Hilbert space (or a reflexive Banach space) has a non-trivial closed invariant subspace. A huge stride towards solving the Invariant Subspace Problem had been made by N. Aronszajn and K. T. Smith [13] in 1954. They proved that every compact operator on a real or complex Banach space has a non-trivial closed invariant subspace. This was a considerable advancement in the sense that it solves the problem for the space of all compact operators which is a very large 1 class of operators. V. Lomonosov [37] later improved this result by replacing an invariant sub- space by a hyperinvariant subspace (a hyperinvariant subspace of an operator T is a subspace which is invariant under every operator commuting with T). He showed that if X is an infinite dimensional Banach space and T is a non­ zero compact operator, then T has a non-trivial closed hyperinvariant subspace. In fact, V. Lomonosov [37] proved more than this. When X is a complex Ba­ nach space and T is not a scalar multiple of the identity and commutes with a non-zero compact operator, then T has a non-trivial closed hyperinvariant subspace. Numerous articles have been written by several authors over the last fifty years regarding the Invariant Subspace Problem.
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