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Operator Commutation Relations

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

  • Book
  • © 1984

Overview

Authors:
  1. Palle E. T. Jørgensen

    1. Department of Mathematics, University of Pennsylvania, Philadelphia, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Robert T. Moore

    1. Department of Mathematics, University of Washington, Seattle, USA

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    You can also search for this author in PubMed   Google Scholar

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

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Table of contents (12 chapters)

  1. Front Matter

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

    1. Front Matter

      Pages 1-2
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    2. Introduction and Survey

      • Palle E. T. Jørgensen, Robert T. Moore
      Pages 3-36

    3. The Finite-Dimensional Commutation Condition

      • 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
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    2. Domain Regularity and Semigroup Commutation Relations

      • Palle E. T. Jørgensen, Robert T. Moore
      Pages 60-76

    3. Invariant-Domain Commutation Theory Applied to the Mass-Splitting Principle

      • 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
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    2. Graph-Density Applied to Resolvent Commutation, and Operational Calculus

      • Palle E. T. Jørgensen, Robert T. Moore
      Pages 108-130

    3. Graph-Density Applied to Semigroup Commutation Relations

      • Palle E. T. Jørgensen, Robert T. Moore
      Pages 131-149

    4. Construction of Globally Semigroup-Invariant C∞ -Domains

      • 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
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    2. Integration of Smooth Operator Lie Algebras

      • Palle E. T. Jørgensen, Robert T. Moore
      Pages 177-193

    3. Exponentiation and Bounded Perturbation of Operator Lie Algebras

      • Palle E. T. Jørgensen, Robert T. Moore
      Pages 194-226

  6. Back Matter

    Pages 227-232
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  7. Lie Algebras of Vector Fields on Manifolds

    1. Front Matter

      Pages 233-239
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    2. Applications of Commutation Theory to Vector-Field Lie Algebras and Sub-Laplacians on Manifolds

      • Palle E. T. Jørgensen, Robert T. Moore
      Pages 240-274

  8. Derivations on Modules of Unbounded Operators with Applications to Partial Differential Operators on Riemann Surfaces

    1. Front Matter

      Pages 275-278
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    2. Rigorous Analysis of Some Commutator Identities for Physical Observables

      • Palle E. T. Jørgensen, Robert T. Moore
      Pages 279-319

  9. Back Matter

    Pages 320-327
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Keywords

About this book

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.

Reviews

`...the reader obtains the impression that there remains much to discover in commutation theory, and this monograph provides both motivation and a guide to the current state of knowledge.'
Mathematical Reviews (1986)

Authors and Affiliations

  • Department of Mathematics, University of Pennsylvania, Philadelphia, USA

    Palle E. T. Jørgensen

  • Department of Mathematics, University of Washington, Seattle, USA

    Robert T. Moore

Bibliographic Information

  • Book Title: Operator Commutation Relations

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

  • Authors: Palle E. T. Jørgensen, Robert T. Moore

  • Series Title: Mathematics and Its Applications

  • DOI: https://doi.org/10.1007/978-94-009-6328-3

  • Publisher: Springer Dordrecht

  • eBook Packages: Springer Book Archive

  • Copyright Information: D. Reidel Publishing Company, Dordrecht, Holland 1984

  • Hardcover ISBN: 978-90-277-1710-8Published: 29 February 1984

  • Softcover ISBN: 978-94-009-6330-6Published: 21 December 2011

  • eBook ISBN: 978-94-009-6328-3Published: 06 December 2012

  • Edition Number: 1

  • Number of Pages: XVIII, 493

  • Topics: Analysis

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