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Representation of certain Linear Operators in Hilbert Space

Published online by Cambridge University Press:  20 November 2018

Bernard Niel Harvey
Bernard Niel Harvey*
Affiliation:
California State College, Long Beach, California University of Toronto, Toronto, Ontario
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In this paper we represent certain linear operators in a space with indefinite metric. Such a space may be a pair (H, B), where H is a separable Hilbert space, B is a bilinear functional on H given by B(x, y) = [Jx, y], [, ] is the Hilbert inner product in H, and J is a bounded linear operator such that J = J* and J2 = I. If T is a linear operator in H, then ‖T‖ is the usual operator norm. The operator J above has two eigenspaces corresponding to the eigenvalues + 1 and –1.

In case the eigenspace in which J induces a positive operator has finite dimension k, a general spectral theory is known and has been developed principally by Pontrjagin [25], Iohvidov and Kreĭn [13], Naĭmark [20], and others.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 1 , February 1971 , pp. 132 - 150
Copyright
Copyright © Canadian Mathematical Society 1971

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