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A BOUND FOR SIMILARITY CONDITION NUMBERS OF UNBOUNDED OPERATORS WITH HILBERT–SCHMIDT HERMITIAN COMPONENTS

Part of: General theory of linear operators Special classes of linear operators

Published online by Cambridge University Press:  12 September 2014

MICHAEL GIL’
MICHAEL GIL’*
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel email [email protected]
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Abstract

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Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ be a linear unbounded operator in a Hilbert space. It is assumed that the resolvent of $H$ is a compact operator and $H-H^*$ is a Hilbert–Schmidt operator. Various integro-differential operators satisfy these conditions. It is shown that $H$ is similar to a normal operator and a sharp bound for the condition number is suggested. We also discuss applications of that bound to spectrum perturbations and operator functions.

Keywords

operatorssimilaritycondition numbersspectrum perturbationsnorm of operator function

MSC classification

Secondary: 47A30: Norms (inequalities, more than one norm, etc.) 47A55: Perturbation theory 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones) 47B10: Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.) 47B40: Spectral operators, decomposable operators, well-bounded operators, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 97 , Issue 3 , December 2014 , pp. 331 - 342
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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