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

We study the rigidity of the orthogonal projections $P_{n}:H^2(\mathbb{D}^2)\rightarrow H_{n}$, where $H^{2}(\mathbb{D}^{2})=\bigoplus_{n=0}^{\infty}H_{n}$ 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 $T_{z}^{*}$. O