Projections and minimal invariant subspaces in the Hardy space over the bidisk

We study the rigidity of the orthogonal projections $P_n:H^2({\mathbb{D}}^2)\to H_n$, where $H^2({\mathbb{D}}^2)={⨁}_{n=0}^{\mathrm{\infty }}{H}_n$ and each $H_n=H^2(z)w^n$ is a reducing subspace for $T_z$, and we derive properties of the minimal invariant subspaces of $T_z^{\ast }$. Our main result establishes that the nonzero projections of a minimal invariant subspace onto the one-variable Hardy space share a common $t_z^{\ast }$-invariant structure. Furthermore, we relate the results obtained to an approach to the invariant subspace problem.