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Essential numerical ranges for linear operator pencils.

Boegli, Sabine and Marletta, Marco (2020) 'Essential numerical ranges for linear operator pencils.', IMA journal of numerical analysis., 40 (4). pp. 2256-2308.


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.

Item Type:Article
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Publisher 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:
Date accepted:31 August 2019
Date deposited:12 December 2019
Date of first online publication:22 November 2019
Date first made open access:22 November 2020

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