Essential normality of quotient modules vs. Hilbert-Schmidtness of submodules in $H^2(\mathbb D^2)$

TL;DR

This paper proves that certain quotient modules in the Hardy space over the bidisk are essentially normal and establishes a link between their essential normality and the Hilbert-Schmidt property of submodules, advancing understanding in operator theory.

Contribution

It demonstrates the essential normality of quotient modules associated with finitely generated submodules containing a distinguished polynomial in $H^2(\mathbb D^2)$, a novel result in this area.

Findings

All such quotient modules are essentially normal.

Essential normality is equivalent to Hilbert-Schmidt property of submodules containing a polynomial.

Finitely generated submodules containing a polynomial are Hilbert-Schmidt.

Abstract

In the present paper, we prove that all the quotient modules in $H^2(\mathbb D^2)$, associated to the finitely generated submodules containing a distinguished homogenous polynomial, are essentially normal, which is the first result on the essential normality of non-algebraic quotient modules in $H^2(\mathbb D^2)$. Moreover, we obtain the equivalence of the essential normality of a quotient module and the Hilbert-Schmidtness of its associated submodule in $H^2(\mathbb D^2)$, in the case that the submodule contains a distinguished homogenous polynomial. As an application, we prove that each finitely generated submodule containing a polynomial is Hilbert-Schmidt, which partially gives an affirmative answer to the conjecture of Yang \cite{Ya3}.