Discrete Wiener-Hopf operators on spaces with Muckenhoupt weight

The discrete Wiener-Hopf operator generated by a function ${\displaystyle a\left(e^{i\theta }\right)}$ with the Fourier series ${\displaystyle {\sum }_{n\in \mathbb{Z}}{a}_n{e}^{\in \theta }}$ is the operator T(a) induced by the Toeplitz matrix ${\displaystyle {\left(a_{j-k}\right)}_{j,k=0}^{\infty }}$ on some weighted sequence space ${\displaystyle l^p({\mathbb{Z}}_{+},w)}$. We assume that w satisfies the Muckenhoupt ${\displaystyle A_p}$ condition and that a is a piecewise continuous function subject to some natural multiplier condition. The last condition is in particular satisfied if a is of bounded variation. Our main result is a Fredholm criterion and an index formula for T(a). It implies that the essential spectrum of T(a) results from the essential range of a by filling in certain horns between the endpoints of each jump. The shape of these horns is determined by the indices of powerlikeness of the weight w.