On Wiener-Hopf operators over linearly ordered diskrete Abelian groups

TL;DR

This paper analyzes Wiener-Hopf operators on discrete linearly ordered Abelian groups, computing their Fredholm index and spectral properties using Fourier transform techniques.

Contribution

It provides explicit formulas for the Fredholm index and spectral analysis of Wiener-Hopf operators in this algebraic setting, extending classical results.

Findings

Computed Fredholm index for Wiener-Hopf operators.

Characterized spectral properties via symbols and Fourier transform.

Extended analysis to discrete linearly ordered Abelian groups.

Abstract

Let $X$ denotes a discrete linearly ordered Abelian group, and let $X_+$ be the positive cone in $X$. In this note we compute the Fredholm index and study spectral properties of Wiener-Hopf operators $W_kg=1_{X_+}(k\ast g)$, $k\in l_2(X_+)$ in the space $l_2(X_+)$ in terms of their symbols $\check k$ where $\check k$ stands for the inverse Fourier transform of $k$.