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.

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