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On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators

Chandler-Wilde, S., Chonchaiya, R. and Lindner, M. (2024) On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators. Journal of Spectral Theory, 14 (2). pp. 719-804. ISSN 1664-0403

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To link to this item DOI: 10.4171/JST/514

Abstract/Summary

In this paper we derive novel families of inclusion sets for the spectrum and pseudospectrum of large classes of bounded linear operators, and establish convergence of particular sequences of these inclusion sets to the spectrum or pseudospectrum, as appropriate. Our results apply, in particular, to bounded linear operators on a separable Hilbert space that, with respect to some orthonormal basis, have a representation as a bi-infinite matrix that is banded or band-dominated. More generally, our results apply in cases where the matrix entries themselves are bounded linear operators on some Banach space. In the scalar matrix entry case we show that our methods, given the input information we assume, lead to a sequence of approximations to the spectrum, each element of which can be computed in finitely many arithmetic operations, so that, with our assumed inputs, the problem of determining the spectrum of a band-dominated operator has solvability complexity index one, in the sense of Ben-Artzi et al. (arXiv:1508.03280, 2020). As a concrete and substantial application, we apply our methods to the determination of the spectra of non-self-adjoint bi-infinite tridiagonal matrices that are pseudoergodic in the sense of Davies (Commun. Math. Phys. 216 (2001) 687-704).

Item Type:Article
Refereed:Yes
Divisions:Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:116730
Publisher:European Mathematical Society

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