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Definitizable Operators on a Krein Space

Published online by Cambridge University Press:  20 November 2018

Petr Zizler
Petr Zizler*
Affiliation:
Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, T2N 1N4, e-mail:zizler@acs.ucalgary.ca
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Abstract

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Let A be a bounded linear operator on a Hilbert space H. Assume that A is selfadjoint in the indefinite inner product defined by a selfadjoint, bounded, invertible linear operator G on H; [x,y] := (Gx,y). In the first part of the paper we define two orders of neutrality for the pair (G, A) and a connection is made with the "types" of numbers in the point and approximate point spectrum of A. The main results of the paper are in the second part and they deal with strong and uniform definitizability of a bounded selfadjoint operator on a Pontrjagin space. They state:

A) Let A be a bounded strongly definitizable operator on a Pontrjagin space ΠK, then A is uniformly definitizable.

B) A bounded selfadjoint operator A on a Pontrjagin space ΠK is uniformly definitizable if and only if all the eigenvalues of A are of definite type and all the nonisolated eigenvalues of A are of positive type.

Some applications to the theory of linear selfadjoint operator pencils are given.

Keywords

47B5047B15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 38 , Issue 4 , 01 December 1995 , pp. 496 - 506
Copyright
Copyright © Canadian Mathematical Society 1995

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