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Commutation Properties of Hilbert Space Operators and Related Topics (Paperback, Softcover reprint of the original 1st ed. 1967)
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Commutation Properties of Hilbert Space Operators and Related Topics (Paperback, Softcover reprint of the original 1st ed. 1967)
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, 36
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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.
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