Abstract. For every inner function $\psi\in H^2(D)$, the Jordan block $S(\psi )$ is the compression of the unilateral shift to the quotient space $H^2(D)\ominus\psi H^2(D)$. On the Hardy space over the bidisk $H^2(D^2)$, the Toeplitz operators $T_{z}$ and $T_{w}$ are unilateral shifts of infinite multiplicity. For every subspace $M\subset H^2(D^2)$ invariant under $T_{z}$ and $T_{w}$, the associated {\it two-variable Jordan block} $S(M):=(S_{z}, S_{w})$ is the compression of the pair $(T_{z}, T_{w})$ to the quotient space $H^2(D^2)\ominus M$. This paper proves that $S(M)$ has no reducing subspace for any $M$, and gives a detailed study of $S_{w}$ when $S_{z}$ is a strict contraction. The one variable Jordan block $S(\psi )$ and the Toeplitz algebra are special cases of the work in this paper.

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

Received May 14, 2002, and in revised form October 25, 2002. (Registered under 2928/2009.)