Schmidt subspaces of block Hankel operators

TL;DR

This paper extends the understanding of Schmidt subspaces of block Hankel operators in vector-valued Hardy spaces, showing they are nearly $S^*$-invariant with finite defect, and offers a new proof of scalar case characterizations.

Contribution

It generalizes the characterization of Schmidt subspaces to vector-valued Hardy spaces and provides an alternative, concise proof for scalar-valued cases.

Findings

Schmidt subspaces in vector-valued Hardy spaces are nearly $S^*$-invariant with finite defect.

A short proof of scalar-valued Hardy space results is provided.

The work complements existing structural results by Gérard and Pushnitski.

Abstract

In scalar-valued Hardy space, the class of Schmidt subspaces for a bounded Hankel operator are closely related to nearly $S^*$-invariant subspaces, as described by G\'{e}rard and Pushnitski. In this article, we prove that these subspaces in the context of vector-valued Hardy spaces are nearly $S^*$-invariant with finite defect in general. As a consequence, we obtain a short proof of the characterization results concerning the Schmidt subspaces in scalar-valued Hardy space in an alternative way. Thus, our work complements the work of G\'{e}rard and Pushnitski regarding the structure of Schmidt subspaces.