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AN UNBOUNDED OPERATOR WITH SPECTRUM IN A STRIP AND MATRIX DIFFERENTIAL OPERATORS

Part of: Special classes of linear operators Operator theory: Ordinary differential operators General theory of linear operators

Published online by Cambridge University Press:  16 April 2021

MICHAEL GIL’ Open the ORCID record for MICHAEL GIL’ [Opens in a new window]
MICHAEL GIL’*
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, PO Box 653, Beer-Sheva 84105, Israel
*
e-mail: [email protected]
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Abstract

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Let A and $\tilde A$ be unbounded linear operators on a Hilbert space. We consider the following problem. Let the spectrum of A lie in some horizontal strip. In which strip does the spectrum of $\tilde A$ lie, if A and $\tilde A$ are sufficiently ‘close’? We derive a sharp bound for the strip containing the spectrum of $\tilde A$ , assuming that $\tilde A-A$ is a bounded operator and A has a bounded Hermitian component. We also discuss applications of our results to regular matrix differential operators.

Keywords

differential operatorHilbert spacespectrum localisation

MSC classification

Primary: 47A10: Spectrum, resolvent
Secondary: 47A55: Perturbation theory 47B10: Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.) 47E05: Ordinary differential operators (should also be assigned at least one other classification number in section 47)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 146 - 153
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Australian Mathematical Publishing Association Inc. 2021

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