A Nehari type theorem and applications to Toeplitz+Hankel operators on the Dirichlet space

Abstract

We study the characterization problem for Hankel operators on the Dirichlet space in relation to Nehari's well-known result for Hankel operators on the Hardy space. We show that the operator equation $T_{\overline{z}}H=H{T}_z$ completely characterizes Hankel operators among all bounded operators on the Dirichlet space where $T_u$ denotes the Toeplitz operator with symbol u. As an application, we obtain a characterization of when a finite sum of products of two Toeplitz+Hankel operators is itself a Toeplitz+Hankel operator.