Infinite-Dimensional Analysis: Operators in Hilbert Space; Stochastic Calculus via Representations, and Duality Theory

253 Pages·2021·19.166 MB·English

by  Palle Jorgensen, James Tian

INFINITE-
DIMENSIONAL
ANALYSIS
Operators in Hilbert Space; Stochastic Calculus
via Representations, and Duality Theory
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ISBN: 978-981-3202-11-5
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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
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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.
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Dedicated to the memory of Ola Bratteli, Richard V. Kadison, and
Edward Nelson.
v
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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
January4,2021 19:57 Infinite-DimensionalAnalysis-9inx6in b4068-fm pageix
Preface ix
associatedinfinite-dimensionalcalculusofItoˆandMalliavin. Ourapproach
is via positive definite kernels, and their associated Reproducing Kernel
Hilbert Spaces (RKHS). In Chapters 6 and 7, we make the connection
between the study of representations of the CCRs, and their use in anal-
ysis of Gaussian fields. This connection will be made with the use of an
important concept from representation theory, intertwining operators.
The last chapter covers such applications as (i) analysis of networks of
resisters,relativereproducingkernels(infinitegraphsofresisters,reversible
Markovprocesses); and (ii) use of RKHS theory for optimizationproblems
in machine learning.
Anticipated audience: “Travel guide” for different groups of readers.
Below, we callattentionto the followingsix overlappinggroups. Theseare
the groups of potential readers the present co-authors have encountered in
their own work/travels. We expect that there may be others as well.
(i) Courses which stress interdisciplinary topics; we expect a variety of
course levels, including departments of mathematics, physics, and
statistics.
(ii) Mathematicians familiar with functional analysis, but not with all of
the diverse applications; including the multi-faceted Hilbert’s Sixth
Problem.
(iii) Physicists familiar with aspects of Hilbert’s Sixth Problem, but who
might be shaky regarding their mathematical underpinnings.
(iv) Mathematicians from anyone of many areas, and levels of sophistica-
tion,whoarecuriousaboutallthe“fancytalk”aboutquantumphysics
and random fields, etc;
(v) Supplementary text for students in beginning graduate courses, or
topic courses (in math or in neighboring departments);
(vi) A general reader looking to broaden her/his perspective.