Journal of Operator Theory

Volume 42, Issue 2, Fall 1999 pp. 379-404.

Berger-Shaw theorem in Hardy module over the bidisk

Authors: Rongwei Yang
Author institution: Department of Mathematics, University of Georgia, Athens, GA 30602, USA

Summary: It is well known that the Hardy space over the bidisk ${\bbb D}^{2}$ is an $A({\bbb D}^2)$ module and that $A({\bbb D}^{2})$ is contained in $\hh$. Suppose $(h)\subset A({\bbb D}^{2})$ is the principal ideal generated by a polynomial $h$, then its closure $[h](\subset \hh)$ and the quotient $\hh \ominus [h]$ are both $A({\bbb D}^{2})$ modules. We let $R_{z},R_{w}$ be the actions of the coordinate functions $z$ and $w$ on $[h]$, and let $S_{z},S_{w}$ be the actions of $z$ and $w$ on $\hh \ominus [h]$. In this paper, we will show that $R_{z}$ and $R_{w}$, as well as $S_{z}$ and $S_{w}$, essentially doubly commute. Moreover, both $[R^{*}_{w},R_{z}]$ and $[S^{*}_{w},S_{z}]$ are actually Hilbert-Schmidt.

Keywords: Bidisk, cross commutator, Hilbert-Schmidt, submodule

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