# Commutation Properties of Hilbert Space Operators and Related Topics

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

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

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.

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

- 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
- About this book