Kernels of Wiener-Hopf plus Hankel operators with matching generating functions

TL;DR

This paper provides an explicit description of the kernels and cokernels of Wiener-Hopf plus Hankel operators with matching generating functions, under certain algebraic conditions, advancing the understanding of their structure.

Contribution

It introduces a new explicit characterization of kernels and cokernels for these operators when their generating functions satisfy a specific matching condition.

Findings

Explicit kernel and cokernel descriptions derived

Conditions under which the descriptions hold identified

Advances the theoretical understanding of Wiener-Hopf plus Hankel operators

Abstract

Considered are Wiener--Hopf plus Hankel operators $W(a)+H(b):L^p(\mathbb{R}^+)\to L^p(\mathbb{R}^+)$ with generating functions $a$ and $b$ from a subalgebra of $L^\infty(\mathbb{R})$ containing almost periodic functions and Fourier images of $L^1(\mathbb{R})$-functions. If the generating functions $a$ and $b$ satisfy the matching condition \begin{equation*} a(t) a(-t)=b(t) b(-t),\quad t\in\mathbb{R}, \end{equation*} an explicit description for the kernels and cokernels of the operators mentioned is given.