# 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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