Journal of Operator Theory

Volume 51, Issue 2, Spring 2004 pp. 361-376.

Backward shift invariant subspaces in the bidisc. II

Authors: Keiji Izuchi, (1) Takahiko Nakazi, (2) and Michio Seto (3)
Author institution: (1) Department of Mathematics, Niigata University, Niigata 950-2181, Japan
(2) Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
(3) Department of Mathematics, Tohoku University, Senndai 980-8578, Japan

Summary: For every invariant subspace

$M$ in the Hardy spaces

$H^2(\Gamma^2)$ , let

$V_z$ and

$V_w$ be multiplication operators on

$M$ . Then it is known that the condition

$V_zV^*_w = V^*_wV_z$ on

$M$ holds if and only if

$M$ is a Beurling type invariant subspace. For a backward shift invariant subspace

$N$ in

$H^2(\Gamma^2)$ , two operators

$S_z$ and

$S_w$ on

$N$ are defined by

$S_z = P_NL_zP_N$ and

$S_w = P_NL_wP_N$ , where

$P_N$ is the orthogonal projection from

$L^2(\Gamma^2)$ onto

$N$ . It is given a characterization of

$N$ satisfying

$S_zS^*_w = S^*_wS_z$ on

$N$ .

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