This paper is concerned with Wiener-Hopf integral operators on L^p and with Toeplitz operators (or matrices) on l^p. The symbols of the operators are assumed to be continuous matrix functions. It is well known that the invertibility of the operator itself and of its associated operator imply the invertibility of all sufficiently large truncations and the uniform boundedness of the norms of their inverses. Quantitative statements, such as results on the limit of the norms of the inverses, can be proved in the case p=2 by means of C*-algebra techniques. In this paper we replace C*-algebra methods by more direct arguments to determine the limit of the norms of the inverses and thus also of the pseudospectra of large truncations in the case of general p.

The research of the first author was supported by the Alfried Krupp Förderpreis für junge Hochschullehrer of the Krupp Foundation. The research of the second author was supported by the Russian fund of fundamental investigations (Grants N 95-01-01285a and N 96-01-01195a).