Essential numerical ranges for linear operator pencils

Bogli, Sabine and Marletta, Marco ORCID: https://orcid.org/0000-0003-1546-4046 2020. Essential numerical ranges for linear operator pencils. IMA Journal of Numerical Analysis 40 (4) , pp. 2256-2308. 10.1093/imanum/drz049

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Abstract

We introduce concepts of essential numerical range for the linear operator pencil λ → A − λB. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem Tx = λx into the pencil problem BTx = λBx for suitable choices of B, we can obtain non-convex spectral enclosures for T and, in the study of truncation and projection methods, confine spectral pollution to smaller sets than with hitherto known concepts. We apply the results to various block operator matrices. In particular, Theorem 4.12 presents substantial improvements over previously known results for Dirac operators while Theorem 4.5 excludes spectral pollution for a class of non-selfadjoint Schro ̈dinger operators which it has not been possible to treat with existing methods.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	Oxford University Press
ISSN:	0272-4979
Funders:	Swiss National Science Foundation
Date of First Compliant Deposit:	4 September 2019
Date of Acceptance:	31 August 2019
Last Modified:	03 Dec 2024 07:45
URI:	https://orca.cardiff.ac.uk/id/eprint/125263

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