Abstract. On the Hardy space over the torus $H^2(\Gamma ^2)$, the Toeplitz operators $T_{z}$ and $T_{w}$ are unilateral shifts of infinite multiplicity. Subspaces $N\subset H^2(\Gamma ^2)$ invariant under $T^*_{z}$ and $T^*_{w}$ are said to be backward shift invariant. This paper studies the compression of the pair $(T_{z}, T_{w})$ (denoted by $(S_{z}, S_{w})$) to $N$. Its focus lies on the case when $S_z$ is a strict contraction. Much information about $N^{\perp }$ can be deduced in this case.

AMS Subject Classification (1991): 46E20, 47A20, 47A13

Received December 30, 2002, and in final form April 3, 2003. (Registered under 5805/2009.)