Operator Commutation Relations: Commutation Relations for Operators, Semigroups, and Resolvents with Applications to Mathematical Physics and Representations of Lie Groups

505 Pages·1984·15.707 MB·English

by  Palle E. T. Jørgensen, Robert T. Moore (auth.)

Operator Commutation Relations
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics a.nd Computer Science, Amsterdam, The Netherlands
Editorial Board:
R. W. BROCKETT, Harvard University, Cambridge, Mass., U.S.A.
J. CORONES, Iowa State University, U.S.A. and Ames Laboratory, U.S. Department of
Energy, Iowa, U.S.A.
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands
G.-C. ROTA,M.lT., Cambridge, Mass., U.S.A.
Operator
Commutation Relations
Commutation Relations for Operators,
Semigroups, and Resolvents with Applications to
Mathematical Physics and Representations
of Lie Groups
Palle E. T. Jorgensen
Department ofM athematics,
University of Pennsylvania, Philadelphia, U.S.A.
and
Robert T. Moore
Department of Mathematics,
University of Washington, Seattle, U.S.A.
D. Reidel Publishing Company
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht I Boston I Lancaster
library of Congress Cataloging in Publication Data
Jt\rgensen, Palle E. T., 1947-
Operator commutation relations.
(Mathematics and its applications)
Bibliography: p.
Includes index.
1. Partial differential operators. 2. Commutation relations (Quantum
mechanics) 3. Lie groups. 4. Representations of groups. I. Moore, Robert T.,
1938- II. Title. III. Series: Mathematics and its applications (D. Reidel
Publishing Company)
QA329.42.J67 1984 515.7'242 83-26957
ISBN-13: 978-94-009-6330-6 e-ISBN -13: 978-94-009-6328-3
DOl: 10.1007/978-94-009-6328-3
Published by D. Reidel Publishing Company,
P.O. Box 17,3300 AA Dordrecht, Holland
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
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In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, Holland
All Rights Reserved
© 1984 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover 1st edition 1984
No part of the material protected by this copyright notice may be reproduced or utilized
in any form or by any means, electronic or mechanical, including photocopying,
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without written permission from the copyright owner.
TABLE OF CONTENTS
Preface ix
Chapter Dependency Diagram xv
Acknowledgements xvii
PART I: SOME MAIN RESULTS ON COMMUTATOR IDENTITIES
Chapter 1. Introduction and Survey 3
1A General Objectives of the Monograph 3
lB Contact with Prior Literature 6
lC The Main Results in Commutation Theory 7
lD The Main Results in Exponentiation Theory 14
lE Results on (Semi) Group-invariant Coo-domains 19
lF Typical Applications of Commutation Theory 23
lG Typical Applications of Exponentiation Theory 29
Chapter 2. The Finite-Dimensional Commutation Condition 37
2A Implications of Finite-Dimensionality in
Commutation Theory 37
2B Examples involving Differential Operators 40
2C Examples from Universal and Operator Enveloping
Algebras 42
2D Relaxing the Finite-Dimensionality Condition 50
PART II: COMMUTATION RELATIONS AND REGULARITY PROPERTIES
FOR OPERATORS IN THE ENVELOPING ALGEBRA OF
REPRESENTATIONS OF LIE GROUPS
Introduction 57
Chapter 3. Domain Regularity and Semigroup Commutation
Relations 60
3A Lie Algebras of Continuous Operators 62
3B Semigroups and Ad-Orbits 64
3C Variations upon the Regularity Condition 67
3D Infinite-DimensionalOA(B) 74
Chapter 4. Invariant-Domain Commutation Theory applied
to the Mass-Splitting Principle 77
4A Global InvariancejRegularity for Heat-Type
Semigroups 79
vi TABLE OF CONTENTS
4B Formulation of the Generalized Mass-Splitting
Theorem 82
4c The Mass-Operator as a Commuting Difference of
Sub-Laplacians 83
4D Remarks on General Minkowskian Observables 91
4E Fourier Transform Calculus and Centrality of
Isolated Projections 95
PART III: CONDITIONS FOR A SYSTEM OF UNBOUNDED OPERATORS
TO SATISFY A GIVEN COMMUTATION RELATION
Introduction 101
Chapter 5. Graph-Density applied to Resolvent
Commutation, and Operational Calculus 108
5A Augmented Spectra and Resolvent Commutation
Relations 111
5B Commutation Relations on Dl 117
5C Analytic Continuation of Commutation Relations 122
5D Commutation Relations for the Holomorphic
Operational Calculus 124
Chapter 6. Graph-Density Applied to Semi group
Commutation Relations 131
6A Semigroup Commutation Relations with a Closable
Basis 132
6B Variants of Sections 5B and 6A for General M 139
6c Automatic Availability of a Closable Basis 144
6D Remarks on Operational Calculi 146
Chapter 7. Construction of Globally Semigroup-invariant
COO-domains 150
7A Frechet COO-domains in Banach Spaces 151
7B The Extrinsic Two-Operator Case 155
7C The Lie Algebra Case 160
7D Coo-action of Resolvents, Projections, and
Operational Calculus 164
PART IV: CONDITIONS FOR A LIE ALGEBRA OF UNBOUNDED
OPERATORS TO GENERATE A STRONGLY CONTINUOUS
REPRESENTATION OF THE LIE GROUP
Introduction 173
Chapter 8. Integration of Smooth Operator Lie Algebras 177
8A Smooth Lie Algebras and Differentiable
Representations 178
8B Applications in Coo-vector spaces 186
Chapter 9. Exponentiation and Bounded Perturbation of
Operator Lie Algebras 194
9A Discussion of Exponentiation Theorems and
Applications 194
TABLE OF CONTENTS vii
9B Proofs of the Theorems 199
9C Phillips Perturbations of Operator Lie Algebras
and Analytic Continuation of Group
Representations 206
9D Semidirect Product Perturbations 217
Appendix to Part IV 227
PART V: LIE ALGEBRAS OF VECTOR FIELDS ON MANIFOLDS
Introduction 235
Chapter 10. Applications of Commutation Theory to
Vector-Field Lie Algebras and Sub-
Lap1acians on Manifolds 240
lOA Exponentials versus Geometric Integrals of
Vector-Field Lie Algebras 243
lOB Exponentiation on LP spaces 251
10C Sub-Laplacians on Manifolds 265
10D Solution Kernels on Manifolds 268
PART VI: DERIVATIONS ON MODULES OF UNBOUNDED OPERATORS
WITH APPLICATIONS TO PARTIAL DIFFERENTIAL
OPERATORS ON RIEMANN SURFACES
Introduction 277
Chapter 11. Rigorous Analysis of Some Commutator
Identities for Physical Observables 279
llA Variations upon the Graph-Density and
Kato Conditions 282
IlB Various forms of Strong Commutativity 287
llC Nilpotent Commutation Relations of Generalized
Heisenberg-Weyl Type 302
Appendix to Part VI 320
PART VII: LIE ALGEBRAS OF UNBOUNDED OPERATORS:
PERTURBATION THEORY, AND ANALYTIC
CONTINUATION OF st(2,lli)-MODULES
Introduction 331
Chapter 12. Exponentiation and Analytic Continuation
of Heisenberg-Matrix Representations for
st(2,lli) 335
12A Connections to the Theory of TCI Representations
of Semisimple Groups on Banach Spaces 339
12B The Graph-Density Condition and Base-Point
Exponentials 355
12C COO-integrals and Smeared Exponentials on tP 365
12D The Operators AO' AI' and A2 376
12E Compact and Phillips Perturbations 389
viii TABLE OF CONTENTS
12F Perturbations and Analytic Continuation of
Smeared Representations 399
12G Irreducibility, Equivalences, ~nitarity, and
Single-Valuedness 407
12H Perturbation and Reduction Properties of
other Analytic Series 423
121 A Counter-Theorem on Group-Invariant Domains 429
Appendix to Part VII 432
GENERAL APPENDICES
Appendix A. The Product Rule for Differentiable
Operator Valued Mappings 439
Appendix B. A Review of Semi group Folklore, and
Integration in Locally Convex Spaces 443
Appendix C. The Square of an Infinitesimal Group
Generator 451
Appendix D. An Algebraic Characterization of 0 2(B) 457
A
Appendix E. Compact Perturbations of Semigroups 461
LP
Appendix F. Numerical Ranges, and Semigroups on
spaces 465
Appendix G. Bounded Elements in Operator Lie Algebras 470
References 476
References to Quotations 486
Index 487
List of Symbols 491
PREFACE
In his Retiring Presidential address, delivered before the
Annual Meeting of The American Mathematical Society on
December, 1948, the late Professor Einar Hille spoke on his
recent results on the Lie theory of semigroups of linear
transformations, ..• "So far only commutative operators have
been considered and the product law ... is the simplest
possible. The non-commutative case has resisted numerous
attacks in the past and it is only a few months ago that
any headway was made with this problem. I shall have the
pleasure of outlining the new theory here; it is a blend of
the classical theory of Lie groups with the recent theory of
one-parameter semigroups." The list of references in the
subsequent publication of Hille's address (Bull. Amer. Math •.
Soc. 56 (1950)) includes pioneering papers of I.E. Segal,
I.M. Gelfand, and K. Yosida. In the following three decades
the subject grew tremendously in vitality, incorporating a
number of different fields of mathematical analysis. Early
papers of V. Bargmann, I.E. Segal, L. G~ding, Harish-Chandra,
I.M. Singer, R. Langlands, B. Konstant, and E. Nelson
developed the theoretical basis for later work in a variety
of different applications: Mathematical physics, astronomy,
partial differential equations, operator algebras, dynamical
systems, geometry, and, most recently, stochastic filtering
theory. As it turned out, of course, the Lie groups, rather
than the semigroups, provided the focus of attention.
However, the operator theoretic viewpoint is the same, and
the influence of Hille's original ideas is present in the
later applications. This book is devoted to some of the
important developments since 1948.
Dedicating our book to the memory of Einar Hille, we
find it quite appropriate to begin with a quote from Kipling:
"And each man hears as the twilight nears, to the beat of his
dying heart, the Devil drum on the darkened pane: 'You did it,
but was it Art?'''. This, in fact, was Hille's own beginning
quote in his first edition of Functional Analysis and Semi
groups.
Although, on the surface, the different topics of the
ix
x PREFACE
present book are quite distinct, we shall show that a certain
analysis of operator commutation relations (or rather a
structure of identities for families of operators, and iterated
commutators of the individual operators) leads to an unexpected
unification, as well as to a variety of new results, in diverse
areas of mathematics and applications.
As a first indication of what is involved, consider the
familiar regularity problem for solutions to partial differ
ential equations. Suppose A is a linear partial differential
operator, and suppose that a fixed linear space of functions,
or sections in a vector bundle, has been chosen such that the
resolvent operator (AI-A)-l provides a solution formula for
the inhomogeneous equation AU - Au = f with given right-hand
f in E. As is well known, the regularity problem for the
solution u can then be stated in terms of domains for the
first order vector fields a/ax. , i=l, ... ,n. One way to check
1
regularity, i.e., to check whether the solution u is in the
domain of ajax. , or in the domain of higher order polynomials
1
in the a/ax. IS, is to consider the formal expansion,
1
Suppose the convergence questions which are implicit in
the series can be resolved, then by simple associative algebra
one would expect the formula to represent the x.-derivative of
1
the solution, viz., au/ax. where
1
au/ax. = (a/ax.)u = (a/ax.)(AI-A)-l(f).
1 1 1
Hence, the regularity question has been reformulated in terms
of a commutation-relation, and a domain problem.
Historically, our work on the subject originated with
a different problem. Rather than the commuting family of
first order operators a/ax., we took, as the starting point,
1
a given Lie algebra ( of operators on a complete normed
space E. What this amounts to is that the elements in ( are
unbounded operators on E, but the different operators have
a common dense invariant domain D in E. The Lie algebra
structure on ( refers to the commutator bracket, i.e.,
elements A and B in ( are considered as operators, and the
=
bracket is.lA,B] AB - BA. Then note that [A,B] is again