Simon Chandler-WildeUniversity of Reading

On Computing the Spectrum and Essential Spectrum of Band-Dominated Operators on the Axis and Half-Axis

Abstract:

In this talk I will describe work, with Marko Lindner and Ratchanikorn Chonchaiya, on deriving convergent sequences of inclusion sets for the spectrum, essential spectrum, and pseudospectrum, of bounded linear operators whose matrix representation is banded or band-dominated. These methods are based on computations of the injection modulus/lower norm of square or rectangular finite sections, using ideas introduced in the self-adjoint case by Davies and Plum (IMA J. Numer. Anal., 2004), and extended to the non-self-adjoint case by Hansen (J. Funct. Anal., 2008). As an application of our general results we consider operators that are pseudo-ergodic in the sense of Davies (Comm. Math. Phys., 2001), for example the Feinberg-Zee random hopping matrix introduced in Feinberg and Zee (Phys. Rev. E, 1999) and studied in Chandler-Wilde and Davies (J. Spectr. Theory, 2012).