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© 1967

Commutation Properties of Hilbert Space Operators and Related Topics

Book

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

  1. 1.Division of Mathematical SciencesPurdue UniversityLafayetteUSA

Bibliographic information

  • Book Title Commutation Properties of Hilbert Space Operators and Related Topics
  • Authors Calvin R. Putnam
  • Series Title Ergebnisse der Mathematik und ihrer Grenzgebiete
  • 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
  • Hardcover ISBN 978-3-540-03778-1
  • Softcover ISBN 978-3-642-85940-3
  • eBook ISBN 978-3-642-85938-0
  • Series ISSN 0071-1136
  • Edition Number 1
  • Number of Pages XII, 168
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Operator Theory
  • Buy this book on publisher's site