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Operator Commutation Relations

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

  • Palle E. T. Jørgensen
  • Robert T. Moore
Book

Part of the Mathematics and Its Applications book series (MAIA, volume 14)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Some Main Results on Commutator Identities

    1. Front Matter
      Pages 1-2
    2. Palle E. T. Jørgensen, Robert T. Moore
      Pages 3-36
    3. Palle E. T. Jørgensen, Robert T. Moore
      Pages 37-53
  3. Commutation Relations and Regularity Properties for Operators in the Enveloping Algebra of Representations of Lie Groups

    1. Front Matter
      Pages 55-59
    2. Palle E. T. Jørgensen, Robert T. Moore
      Pages 60-76
    3. Palle E. T. Jørgensen, Robert T. Moore
      Pages 77-97
  4. Conditions for a System of Unbounded Operators to Satisfy a Given Commutation Relation

    1. Front Matter
      Pages 99-107
    2. Palle E. T. Jørgensen, Robert T. Moore
      Pages 108-130
    3. Palle E. T. Jørgensen, Robert T. Moore
      Pages 131-149
    4. Palle E. T. Jørgensen, Robert T. Moore
      Pages 150-169
  5. Conditions for a Lie Algebra of Unbounded Operators to Generate a Strongly Continuous Representation of the Lie Group

    1. Front Matter
      Pages 171-176
    2. Palle E. T. Jørgensen, Robert T. Moore
      Pages 177-193
    3. Palle E. T. Jørgensen, Robert T. Moore
      Pages 194-226
    4. Back Matter
      Pages 227-232
  6. Lie Algebras of Vector Fields on Manifolds

    1. Front Matter
      Pages 233-239
  7. Derivations on Modules of Unbounded Operators with Applications to Partial Differential Operators on Riemann Surfaces

    1. Front Matter
      Pages 275-278
    2. Palle E. T. Jørgensen, Robert T. Moore
      Pages 279-319
    3. Back Matter
      Pages 320-327
  8. Lie Algebras of Unbounded Operators: Perturbation Theory, and Analytic Continuation of sℓ(2,ℝ)- Modules

    1. Front Matter
      Pages 329-334
    2. Back Matter
      Pages 432-436
  9. Back Matter
    Pages 437-493

About this book

Introduction

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.

Keywords

Riemann surface banach spaces calculus differential equation dynamical systems filtering geometry integral integration manifold mathematical analysis mathematical physics operator operator algebra partial differential equation

Authors and affiliations

  • Palle E. T. Jørgensen
    • 1
  • Robert T. Moore
    • 2
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-009-6328-3
  • Copyright Information Springer Science+Business Media B.V. 1984
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-94-009-6330-6
  • Online ISBN 978-94-009-6328-3
  • Buy this book on publisher's site
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