Compactness of products of block Hankel and Toeplitz operators

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.