Infinite-dimensional Analysis. Operators in Hilbert Space; Stochastic Calculus via Representations, and Duality Theory PDF
Preview Infinite-dimensional Analysis. Operators in Hilbert Space; Stochastic Calculus via Representations, and Duality Theory
INFINITE- DIMENSIONAL ANALYSIS Operators in Hilbert Space; Stochastic Calculus via Representations, and Duality Theory 1111998800__99778899881111222255777722__TTPP..iinndddd 11 3311//1122//2200 1111::3377 AAMM Other World Scientific Titles by the Author Recent Advances in Computational Sciences: Selected Papers from the International Workshop on Computational Sciences and Its Education ISBN: 978-981-270-700-0 Non-commutative Analysis ISBN: 978-981-3202-11-5 ISBN: 978-981-3202-12-2 (pbk) YYuummeenngg -- 1111998800 -- IInnffiinniittee--DDiimmeennssiioonnaall AAnnaallyyssiiss..iinndddd 11 3300//1122//22002200 88::2288::2200 ppmm INFINITE- DIMENSIONAL ANALYSIS Operators in Hilbert Space; Stochastic Calculus via Representations, and Duality Theory Palle Jorgensen The University of Iowa, USA James Tian American Mathematics Society, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO 1111998800__99778899881111222255777722__TTPP..iinndddd 22 3311//1122//2200 1111::3377 AAMM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Jørgensen, Palle E. T., 1947– author. | Tian, James F., author. Title: Infinite-dimensional analysis : operators in Hilbert space ; stochastic calculus via representations, and duality theory / Palle Jorgensen, The University of Iowa, USA, James Tian, American Mathematics Society, USA. Description: New Jersey : World Scientific, [2021] | Includes bibliographical references and index. Identifiers: LCCN 2020041084 | ISBN 9789811225772 (hardcover) | ISBN 9789811225789 (ebook) | ISBN 9789811225796 (ebook other) Subjects: LCSH: Functional analysis. | Hilbert space. | Stochastic analysis. Classification: LCC QA320 .J637 2021 | DDC 515/.7--dc23 LC record available at https://lccn.loc.gov/2020041084 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11980#t=suppl Desk Editor: Liu Yumeng Typeset by Stallion Press Email: [email protected] Printed in Singapore YYuummeenngg -- 1111998800 -- IInnffiinniittee--DDiimmeennssiioonnaall AAnnaallyyssiiss..iinndddd 22 3300//1122//22002200 88::2288::2200 ppmm January4,2021 19:57 Infinite-DimensionalAnalysis-9inx6in b4068-fm pagev Dedicated to the memory of Ola Bratteli, Richard V. Kadison, and Edward Nelson. v b2530 International Strategic Relations and China’s National Security: World at the Crossroads TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk b2530_FM.indd 6 01-Sep-16 11:03:06 AM January4,2021 19:57 Infinite-DimensionalAnalysis-9inx6in b4068-fm pagevii Preface The present exposition combines a variety of themes from infinite- dimensional analysis of special relevance to applications, both in pure mathematics, as well as in related areas. The general theme we have in mind here is often referred to as infinite-dimensional calculus, or infinite- dimensional calculus of variation. Since it entails diverse topics from anal- ysis, we have found that students will often have difficulties getting off the ground when getting started in the field; for example with a search of the current journal literature. We have even found that our current view is missing from many (most) books in the area. We believe that there is therefore a need for such an exposition, cover- ing and combining the relevant areas of analysis. Such a need has become especially apparentin view ofrecent anddiverse trends in topics fromrep- resentation theory, in spectral theory; and their use in such neighboring areasasquantumphysics,andinthestudy ofstochasticprocesses. Addto that such applications as machine learning, neural network, and stochastic optimization. The purpose of this book is to make available to beginning graduate students, and to others, building up from the ground, some core areas of analysis which often are prerequisites for these developments. We begin with a presentation (Chapters 1 and 2) of a selection of topics from the theory of operators in Hilbert space, algebras of operators, and their cor- responding spectral theory; all themes of special relevancefor our purpose. We then turn to a systematic study of a selection of non-commutative themes; again of special relevance to the particular representation theory we will be needing later. For this task, we begin (in Chapter 3) with a study of representations of the canonical commutation relations (CCRs); vii January4,2021 19:57 Infinite-DimensionalAnalysis-9inx6in b4068-fm pageviii viii Infinite-Dimensional Analysis with an emphasis on the requirements of infinite-dimensional calculus of variations, often referred to as Itˆo and Malliavin calculus, Chapters 4–6. We have included a number of features which we hope will make the book more accessible. These should also serve to make the presentation of some technical points inside the book more reader-friendly. For exam- ple, there are places where we flesh out, and articulate in more detail, the big picture. This is done with addition of reader-guides, and a few math- ematical subsections with explanation of interconnections, add historical pointers,aswellaspointerstodiverseapplications. Thisshouldhelpmoti- vate better some of the technical mathematical themes that are perhaps more difficult. Overall, the book is comprised of three main areas, and we have aimedto makeit as painless as possible for readersto transitionfrom one to the next. Organization of the book: A bird’s eye view, and tips for the reader. Whilethechapter-themesinthebookfollowalogicalprogression,some readers might want first to pick selected topics of personal interest. For this reason, we strive to make each individual chapter reasonably self- contained. Some chapters, andsections, canstand on their own, for exam- ple, Sections 1.3 Connection to Quantum Mechanics; 4.4 Gaussian Hilbert space; and 5.1 analysis with Malliavin derivatives. To further help readers navigate the separate topics, we have included multiple tables which serve to summarize main ideas. This list includes Table 1.1: Lattices of projectionsin Hilbert space; Table 4.1: The positive kernel for Brownian motion, and its RKHS analysis. We cite the literature, as needed, inside the book, and at the end of each chapter we include a brief Guide to the literature. Throughout the book we have credited main results to the founders, by name, of the subject at hand. For the benefit novice readers, we have included in the Appendix a list of short biographical sketches. In Chapter 1 we present the part of operator theory, covering bounded and unbounded linear transformations, stressing a systematic account of spectral theory. Because of the needs of operators in the study of repre- sentations of Lie algebras and of the Canonical Commutation Relations (CCR), theemphasishereis onunboundedoperators. Inourpresentation, westressconnectionstoquantumphysics,butaphysicsbackgroundonthe part of readers is not assumed. The material we need from representation theory is covered in Chapters 2 and 3. Chapters 4 and 5 cover stochastic analysis, and the
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