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Multiple Operator Integrals: Development and Applications

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Multiple Operator Integrals: Development and Applications Multiple Operator Integrals: Development and Applications by Anna Tomskova A thesis submitted for the degree of Doctor of Philosophy at the University of New South Wales. School of Mathematics and Statistics Faculty of Science May 2017 PLEASE TYPE THE UNIVERSITY OF NEW SOUTH WALES Thesis/Dissertation Sheet Surname or Family name: Tomskova First name: Anna Other name/s: Abbreviation for degree as given in the University calendar: PhD School: School of Mathematics and Statistics Faculty: Faculty of Science Title: Multiple Operator Integrals: Development and Applications Abstract 350 words maximum: (PLEASE TYPE) Double operator integrals, originally introduced by Y.L. Daletskii and S.G. Krein in 1956, have become an indispensable tool in perturbation and scattering theory. Such an operator integral is a special mapping defined on the space of all bounded linear operators on a Hilbert space or, when it makes sense, on some operator ideal. Throughout the last 60 years the double and multiple operator integration theory has been greatly expanded in different directions and several definitions of operator integrals have been introduced reflecting the nature of a particular problem under investigation. The present thesis develops multiple operator integration theory and demonstrates how this theory applies to solving of several deep problems in Noncommutative Analysis. The first part of the thesis considers double operator integrals. Here we present the key definitions and prove several important properties of this mapping. In addition, we give a solution of the Arazy conjecture, which was made by J. Arazy in 1982. In this part we also discuss the theory in the setting of Banach spaces and, as an application, we study the operator Lipschitz estimate problem in the space of all bounded linear operators on classical Lp-spaces of scalar sequences. The second part of the thesis develops important aspects of multiple operator integration theory. Here, we demonstrate how this theory applies to a solution of the problem on a Koplienko-Neidhardt trace formulae for a Taylor remainder of order two, which was raised by V. Peller in 2005, and also extend the solution for a Taylor remainder of an arbitrary order. Finally, using the tools from multiple operator integration theory, we present an affirmative solution of a question concerning Frechet differentiability of the norm of Lp-spaces, which has been of interest to experts in Banach space geometry for the last 50 years. We resolve this question in the most general setting, namely for the non-commutative Lp-spaces associated with an arbitrary von Neumann algebra, thus answering the open question suggested by G. Pisier and Q. Xu in their influential survey on the geometry of such spaces. Declaration relating to disposition of project thesis/dissertation I hereby grant to the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or in part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all property rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the 350 word abstract of my thesis in Dissertation Abstracts International (this is applicable to doctoral theses only). …………………………………………………………… ……………………………………..……………… ……….……………………...…….… Signature Witness Signature Date The University recognises that there may be exceptional circumstances requiring restrictions on copying or conditions on use. Requests for restriction for a period of up to 2 years must be made in writing. Requests for a longer period of restriction may be considered in exceptional circumstances and require the approval of the Dean of Graduate Research. FOR OFFICE USE ONLY Date of completion of requirements for Award: ORIGINALITY STATEMENT ‘I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.’ Signed …………………………………………….............. Date …………………………………………….............. COPYRIGHT STATEMENT ‘I hereby grant the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the 350 word abstract of my thesis in Dissertation Abstract International (this is applicable to doctoral theses only). I have either used no substantial portions of copyright material in my thesis or I have obtained permission to use copyright material; where permission has not been granted I have applied/will apply for a partial restriction of the digital copy of my thesis or dissertation.' Signed ……………………………………………........................... Date ……………………………………………........................... AUTHENTICITY STATEMENT ‘I certify that the Library deposit digital copy is a direct equivalent of the final officially approved version of my thesis. No emendation of content has occurred and if there are any minor variations in formatting, they are the result of the conversion to digital format.’ Signed ……………………………………………........................... Date ……………………………………………........................... Acknowledgements My deepest gratitude goes to my supervisor Fedor Sukochev first of all for accepting me to his extraordinarily strong mathematical team. I always feel that he believes in my talent more than I do and he inspired me to work on long- standing interesting problems. In addition, he opened for me the mathematical world introducing me to many highly qualified mathematicians and giving me a chance to participate in a series of mathematical conferences and workshops all over the world. My deepest thanks extend to my co-supervisor Denis Potapov who has been very generous with his help and time over these three years and has displayed an astounding level of patience. I would also like to thank my colleague and co-author Dima Zanin for his patience, for sharing his thoughts and allowing me to pester him with questions so often over the years. In addition, my thanks extends to all my co-authors Jan Rozendaal, Christian Le Merdy, Clement Coine and Anna Skripka for a pleasant time of working together. Especially, I would like to thank Christian Le Merdy for the organization of visits to University de Franche-comte, Besanson, France during my study and his great hospitality there. i Contents 1 Overview 2 1.1 Introduction . 2 1.2 Structure of the thesis . 15 2 Double operator integration 18 2.1 Double operator integral on S2 ................... 18 2.2 Double operator integral on B(H) and Schatten classes . 26 2.3 Double operator integral on B(X ; Y) and Banach ideals . 37 3 Arazy conjecture concerning Schur multipliers on Schatten ide- als 50 4 Commutator and Lipschitz estimates in B(X ; Y) 60 4.1 General estimates . 60 4.2 Spaces with unconditional basis . 63 4.3 Absolute value function on B(`p; `q) . 69 4.3.1 Case p < q ........................... 69 4.3.2 Case p ≥ q ........................... 72 4.4 Lipschitz estimates on the ideal of p-summing operators . 74 5 Multiple operator integration 77 5.1 Multiple operator integrals in the finite dimensional setting . 78 5.1.1 Multilinear Schur multipliers via linear ones. 80 5.1.2 Schur multipliers via multiple operator integrals. 83 5.1.3 Properties of multiple operator integrals . 84 ii 5.2 Multiple operator integrals in the setting of non-commutative Lp- spaces . 96 5.2.1 Classical non-commutative Lp-spaces and weak Lp-spaces . 99 5.2.2 Haagerup's Lp-spaces . 102 5.2.3 Multiple operator integrals . 105 5.2.4 H¨older-type estimate . 116 6 Applications to operator Taylor remainders 128 6.1 Affirmative results for Taylor remainders . 129 6.2 Counterexamples for Taylor remainders. 132 6.2.1 Self-adjoint case. 133 6.2.2 Unitary case. 147 7 Differentiation of the norm of Haagerup Lp-space 154 7.1 Traces on L1;1(N ; τ) . 155 7.2 First Fr´echet derivative . 158 7.3 Main estimate . 160 p p 7.4 Taylor expansion for A 7! kAkLp ;A 2 L : . 166 Bibliography 172 1 Chapter 1 Overview 1.1 Introduction Let H be a separable complex Hilbert space. Let B(H) be the spaces of all bounded linear operators on H equipped with the uniform norm k · k1: Let us first consider the finite dimensional situation. Throughout the thesis N = f1; 2;:::g is the set of natural numbers, R and C 2 N H 2 are the sets of real and complex numbers respectively. Let d and = `d, 2 where `d denote the d-dimensional Hilbert space equipped with a scalar product · · 2 ( ; ). Let two self-adjoint linear operators A and B on `d with respective complete f gd f gd f gd systems of orthonormal eigenvectors ξj j=1; ηk k=1 and eigenvalues λj j=1; f gd µk k=1 be given. Let Pξ denote the orthogonal projection on the unit vector 2 2 · · ξ `d; that is, Pξ( ) = ( ; ξ)ξ.
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