On Spectral Inclusion Sets and Computing the Spectra and Pseudospectra of Bounded Linear Operators

TL;DR

This paper introduces new inclusion sets for the spectrum and pseudospectrum of bounded linear operators, demonstrating convergence of these sets to the actual spectra and applying the results to complex matrix classes, including bi-infinite tridiagonal matrices.

Contribution

It develops novel spectral inclusion sets with proven convergence properties and applies these to compute spectra of complex operators with finite arithmetic operations.

Findings

New spectral inclusion sets converge to the spectrum or pseudospectrum.

The methods enable finite-approximation of spectra with solvability index one.

Application to spectra of non-self-adjoint bi-infinite tridiagonal matrices.

Abstract

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. (C. R. Acad. Sci. Paris, Ser. I 353 (2015), 931-936). 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).