Introduction to Scalar Wiener-Hopf Operators

Abstract

Wiener-Hopf operators are convolution operators on a half-line. There is a fascinating interplay between the operator theoretic properties of Wiener-Hopf equations and the topological characteristics of their symbol. This connection is visually perfect for continuous and piecewise continuous symbols. Symbols in C+H∞ which will repeatedly emerge in the forthcoming chapters when studying semi-almost periodic symbols require some more machinery. After proving Bohr’s theorem the notion of the mean motion of an invertible almost periodic function is available. Having the concept of the mean motion we can establish Fredholm and invertibility criteria for Wiener-Hopf operators with almost periodic symbols. In summary this chapter contains the Fredholm theory of scalar Wiener-Hopf operators with symbols in C + H∞ PC and AP.