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Essential numerical ranges for linear operator pencils

Boegli, Sabine; Marletta, Marco

Essential numerical ranges for linear operator pencils Thumbnail


Authors

Marco Marletta



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 nonconvex 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 nonselfadjoint Schrödinger operators which has not been possible to treat with existing methods.

Citation

Boegli, S., & Marletta, M. (2020). Essential numerical ranges for linear operator pencils. IMA Journal of Numerical Analysis, 40(4), 2256-2308. https://doi.org/10.1093/imanum/drz049

Journal Article Type Article
Acceptance Date Aug 31, 2019
Online Publication Date Nov 22, 2019
Publication Date 2020-10
Deposit Date Dec 11, 2019
Publicly Available Date Nov 22, 2020
Journal IMA Journal of Numerical Analysis
Print ISSN 0272-4979
Electronic ISSN 1464-3642
Publisher Oxford University Press
Peer Reviewed Peer Reviewed
Volume 40
Issue 4
Pages 2256-2308
DOI https://doi.org/10.1093/imanum/drz049

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Copyright Statement
This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA journal of numerical analysis following peer review. The version of record 
Boegli, Sabine & Marletta, Marco (2020). Essential numerical ranges for linear operator pencils. IMA Journal of Numerical Analysis 40(4): 2256-2308 is available online at: https://doi.org/10.1093/imanum/drz049




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