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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:[email protected]
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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

References

1. Bognar, J., Indefinite Inner Product Spaces, Springer Verlag, New York, 1974.Google Scholar
2. Conway, J. B., A course in Functional Analysis, Second edition, Springer Verlag, 1989.Google Scholar
3. Dijksma, A. and Gheondea, A., On the Signatures of Selfadjoint Pencils, Dept. of Math., University of Groningen, W-9313, preprint.Google Scholar
4. Jonas, P. and Langer, H.. Compact perturbations of definitizable operators, J. Operator Theory 2(1979), 63–77.Google Scholar
5. Lancaster, P., Markus, A. S. and Matsaev, V. I., Definitizable Operators and Quasihyperbolic Operator polynomials, J. Funct. Anal., to appear.Google Scholar
6. Lancaster, P., Markus, A. S. and Qiang Ye, Low Rank Perturbations of Strongly Definitizable Transformations and Matrix Polynomials, Linear Algebra Appl. 197/198(1994), 3–29.Google Scholar
7. Lancaster, P. and Rodman, L., Minimal symmetric factorizations of symmetric real and complex rational matrix functions, Linear Algebra. Appl., to appear.Google Scholar
8. Lancaster, P., Shkalikov, A. and Qiang Ye, Strongly definitizable linear pencils in Hilbert Space, Integral Equations Operator Theory 17(1993), 338–360.Google Scholar
9. Lancaster, P. and Ye, Qiang, Definitizable hermitian matrix pencils, Aequationes Math. 46( 1993), 44—55.Google Scholar
10. Langer, H., Spectral functions of definitizable operators in Krein spaces, Springer Verlag. Lecture Notes in Math. 948(1982), 1–46.Google Scholar
11. Taylor, A. E. and Lay, D. C., Introduction to Functional Analysis, Second edition, John Wiley and Sons, Inc., 1979.Google Scholar
12. Tomczak, N.-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Pitman Monographs and Surveys Pure Appl. Math., (1987).Google Scholar