Closed form solution of non-homogeneous equations with Toeplitz plus Hankel operators

TL;DR

This paper presents a closed-form solution method for non-homogeneous equations involving Toeplitz plus Hankel operators under a specific matching condition, utilizing Wiener--Hopf factorization to find all solutions.

Contribution

It introduces an efficient solution technique for such equations based on Wiener--Hopf factorization, expanding the analytical tools for Toeplitz plus Hankel operator equations.

Findings

Provides explicit solutions under the matching condition

Utilizes Wiener--Hopf factorization for solution derivation

Enables comprehensive solution characterization

Abstract

Considered is the equation $$ (T(a)+H(b))\phi=f, $$ where $T(a)$ and $H(b)$, $a,b\in L^\infty(\mathbb{T})$ are, respectively, Toeplitz and Hankel operators acting on the classical Hardy spaces $H^p(\mathbb{T})$, $1<p<\infty$. If the generating functions $a$ and $b$ satisfy the so-called matching condition [1,2], $$ a(t) a(1/t)=b(t)b(1/t), \, t\in \mathbb{T}, $$ an efficient method for solving equations with Toeplitz plus Hankel operators is proposed. The method is based on the Wiener--Hopf factorization of the scalar functions $c(t)=a(t)b^{-1}(t)$ and $d(t)=a(t)b^{-1}(1/t)$ and allows one to find all solutions of the equations mentioned.