Compactness of products of block Hankel and Toeplitz operators

TL;DR

This work addresses the compactness of products $H_{\Phi}T_{\Psi}$ of block Hankel and block Toeplitz operators on the vector-valued Hardy space $H_E^2$, motivated by Sarason’s Toeplitz-product problems. It develops a harmonic-extension framework combined with Douglas algebras and maximal ideal-space localization to derive a complete set of equivalent conditions for compactness, including explicit trace-based criteria and local boundary conditions on the maximal ideal space. A central achievement is a precise characterization of when $H_{\Phi}T_{\Psi}$ is itself a block Hankel operator, expressed via a finite-rank matrix condition $\Phi(z)(I-A), A\Psi(z)\in H_{B(E)}^{\infty}$ and $H_{\Phi}T_{\Psi}=H_{\Phi A\Psi}$. The paper then proves a comprehensive set of equivalences (Theorem maina) for compactness, and demonstrates applications such as recovering known scalar results and highlighting the complexity of the matrix-valued case through concrete corollaries.

Abstract

Motivated by the Sarason problem on the products of Hankel and Toeplitz operators on analytic function spaces, we characterize the compactness of products of block Hankel and Toeplitz operators on the vector-valued Hardy space of the unit disk via harmonic extension of the symbols and Douglas algebras generated by the symbols. Additionally, we provide a complete answer to the question of when the product of a block Hankel operator and a block Toeplitz operator is a block Hankel operator.

Theorems & Definitions (47)

• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• Corollary 2.2
• Definition 2.3
• Lemma 2.4