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Commutation Properties of Hilbert Space Operators and Related Topics

  • C. R. Putnam

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 36)

Table of contents

  1. Front Matter
    Pages I-XI
  2. C. R. Putnam
    Pages 1-14
  3. C. R. Putnam
    Pages 15-41
  4. C. R. Putnam
    Pages 42-62
  5. Back Matter
    Pages 147-167

About this book

Introduction

What could be regarded as the beginning of a theory of commutators AB - BA of operators A and B on a Hilbert space, considered as a dis­ cipline in itself, goes back at least to the two papers of Weyl [3] {1928} and von Neumann [2] {1931} on quantum mechanics and the commuta­ tion relations occurring there. Here A and B were unbounded self-adjoint operators satisfying the relation AB - BA = iI, in some appropriate sense, and the problem was that of establishing the essential uniqueness of the pair A and B. The study of commutators of bounded operators on a Hilbert space has a more recent origin, which can probably be pinpointed as the paper of Wintner [6] {1947}. An investigation of a few related topics in the subject is the main concern of this brief monograph. The ensuing work considers commuting or "almost" commuting quantities A and B, usually bounded or unbounded operators on a Hilbert space, but occasionally regarded as elements of some normed space. An attempt is made to stress the role of the commutator AB - BA, and to investigate its properties, as well as those of its components A and B when the latter are subject to various restrictions. Some applica­ tions of the results obtained are made to quantum mechanics, perturba­ tion theory, Laurent and Toeplitz operators, singular integral trans­ formations, and Jacobi matrices.

Keywords

Hilbert space Hilbertscher Raum Jacobi Operators commutation differential operator equation form integral matrices mechanics perturbation perturbation theory spectral theory theorem

Authors and affiliations

  • C. R. Putnam
    • 1
  1. 1.Division of Mathematical SciencesPurdue UniversityLafayetteUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-85938-0
  • Copyright Information Springer-Verlag Berlin Heidelberg 1967
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-85940-3
  • Online ISBN 978-3-642-85938-0
  • Series Print ISSN 0071-1136
  • Buy this book on publisher's site