Representations of Groups and Algebras in Spaces with Indefinite Metric

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− = P−H. 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−).