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

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.