On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators
Journal article
Authors/Editors
Strategic Research Themes
Publication Details
Author list: Simon Chandler-Wilde, Ratchanikorn Chonchaiya, Marko Lindner
Publisher: European Mathematical Society Press
Publication year: 2024
Volume number: 14
Start page: 719
End page: 804
Number of pages: 86
ISSN: 1664-039X
eISSN: 1664-0403
URL: https://ems.press/content/serial-article-files/47828
Languages: English-United States (EN-US)
Abstract
In this paper, we derive novel families of inclusion sets for the spectra and pseudospectra 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. (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 pseudo ergodic in the sense of Davies [Commun. Math. Phys. 216 (2001), 687–704].
Keywords
band-dominated matrix, band matrix, pseudoergodic, solvability complexity index