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Representations of Groups and Algebras in Spaces with Indefinite Metric

  • M. A. Naimark
  • R. S. Ismagilov
Part of the Progress in Mathematics book series (PM, volume 10)

Abstract

Let H be a Hilbert space with scalar product [x, y], and let P+, P be orthogonal projectors in H, where P+ + P = E; we set J + P+ − P, H+ = P+ H, and H = PH. In H we defined the bilinear form
(1)
which we call the indefinite scalar product. If k = min (dim H+, dim H), then H [assigned form (1)] is said to be a space of type Πk, a well as a J-space; we also say that form (1) determines in H the Πk-metric, or indefinite metric. We shall have frequent occasion to denote the space H by Πk. If k < ∞, then Πk is called a Pontryagin space (after L. S. Pontryagin). In this case we shall assume that k = dim H+ (this can always be accomplished, of course, by commutation of the operators P+ and P).

Keywords

Hilbert Space Invariant Subspace Unitary Representation Lorentz Group Symmetric Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1971

Authors and Affiliations

  • M. A. Naimark
  • R. S. Ismagilov

There are no affiliations available

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