Inner sequence based invariant subspaces in $H^{2}(D^2)$

A closed subspace $H^{2}(D^2)$ is said to be invariant if it is invariant under the Toeplitz operators $T_z$ and $T_w$. Invariant subspaces of $H^{2}(D^2)$ are well-known to be very complicated. So discovering some good examples of invariant subspaces will be beneficial to the general study. This paper studies a type of invariant subspace constructed through a sequence of inner functions. It will be shown that this type of invariant subspace has direct connections with the Jordan operator. Related calculations also give rise to a simple upper bound for $\sum_j 1-\vert\lambda_j\vert$, where $\{\lambda_j\}$ are zeros of a Blaschke product.