General Wiener-Hopf operators

Abstract

We call any operator of the form (1.3) a Wiener-Hopf operator or, indifferently, a Toeplitz operator.When we wish to emphasize that we are discussing (1.3) with no restriction on £ or on A, we call (1.3) a general Wiener-Hopf operator.To distinguish the special case (1.2), we call it the special Wiener-Hopf operator.There is a large literature on questions relating to the invertibility of the special Wiener-Hopf operator.It turns out that many of the most important results proved in the literature can be extended to apply also to general Wiener-Hopf operators.This was exploited by Shinbrot in [12] and [13] when A is a positive operator.It is our purpose in this paper to continue the analysis begun in [12] and [13], omitting the hypothesis that A is positive.To see the type of result we are able to get at, we state here a necessary and sufficient condition for the invertibility of £p(A) due to Devinatz ([3]; but see also [17]).Let 77e0 denote the set of all essentially bounded elements of H2.Then, the result of [3] that we have in mind is this.Suppose that (1.4) ess inf |a(0)| > 0.