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

Abstract

Considered is the equation $$(T(a)+H(b))\varphi =f,$$ 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.

18 pages