Wiener-Hopf plus Hankel operators: Invertibility Problems

TL;DR

This paper investigates the invertibility conditions of Wiener-Hopf plus Hankel operators on $L^p$ spaces, providing necessary and sufficient criteria and explicit inverse representations under certain algebraic conditions.

Contribution

It establishes new necessary and sufficient conditions for invertibility of these operators and offers explicit inverse formulas when specific algebraic relations hold.

Findings

Necessary and sufficient invertibility conditions derived.

Explicit inverse representations provided.

Conditions involving algebraic relations between symbols.

Abstract

The invertibility of Wiener-Hopf plus Hankel operators $W(a)+H(b)$ acting on the spaces $L^p(\mathbb{R}^+)$, $1 < p<\infty$ is studied. If $a$ and $b$ belong to a subalgebra of $L^\infty(\mathbb{R})$ and satisfy the condition \begin{equation*} a(t) a(-t)=b(t) b(-t),\quad t\in\mathbb{R}, \end{equation*} we establish necessary and also sufficient conditions for the operators $W(a)+H(b)$ to be one-sided invertible, invertible or generalized invertible. Besides, efficient representations for the corresponding inverses are given.