Compactness of products of block Hankel and Toeplitz operators

TL;DR

This paper characterizes when products of block Hankel and Toeplitz operators are compact on vector-valued Hardy spaces, using harmonic extensions and Douglas algebras, and clarifies conditions for their product to be a block Hankel operator.

Contribution

It provides a complete characterization of the compactness of such operator products and answers when their product remains a block Hankel operator, advancing operator theory.

Findings

Characterization of compactness via harmonic extension and Douglas algebras

Conditions when product of block Hankel and Toeplitz is a block Hankel operator

Complete solution to Sarason problem in this context

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.