Abstracts Category : Other

Multiple operator integrals: development and applications

by Anna Tomskova

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 severaldeep problems in Noncommutative Analysis.The first part of the thesis considers double operator integrals. Here we present the key definitions and prove several importantproperties of this mapping. In addition, we give a solution of the Arazy conjecture, which was made by J. Arazy in 1982. In this partwe also discuss the theory in the setting of Banach spaces and, as an application, we study the operator Lipschitz estimateproblem 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 wasraised 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.Advisors/Committee Members: Sukochev, Fedor, Mathematics & Statistics, Faculty of Science, UNSW, Potapov, Denis, Mathematics & Statistics, Faculty of Science, UNSW.