###### Options

# Banach algebra techniques to compute spectra, pseudospectra and condition spectra of some block operators with continuous symbols

Date Issued

01-01-2015

Author(s)

Krishna Kumar, G.

Kulkarni, S. H.

Abstract

In this paper we use Banach algebra techniques to study the spectrum, pseudospectrum and condition spectrum of a block Laurent operator with continuous symbol and a lower triangular block Toeplitz operator with continuous symbol. (1) Let L be a block Laurent operator with a continuous symbol f. Regarding f as an element of the Banach algebra of all continuous matrix valued functions defined on the unit circle Γ, we show that the spectrum σ(L) of L coincides with the spectrum σ(f) of f. It is also shown that the spectrum σ(f) can be expressed as a union of the spectra of matrices f(x). Thus Similar results are proved about pseudospectrum for ε > 0 and condition spectrum σε(L) = σε(f) for 0 < ε < 1. (2) Let T be an upper or lower triangular block Toeplitz operator with continuous symbol f. Then f is a continuous matrix valued function defined on the closed unit disc and f is analytic in the open unit disc. It is proved that a similar description can be given about the spectrum σ(T), pseudospectrum Λε(T) for ε > 0 and condition spectrum σε(T) for < ε < 1. These results are illustrated with examples and pictures using Matlab.

Volume

6