Certain invariant subspace structure of $L^2(\Bbb T^2)$ II

Abstract

Let $\mathfrak M$ be an invariant subspace of $L^2(\mathbb{T}^2)$. Considering the largest z-invariant (resp. $w$-invariant) subspace $\mathfrak {F}_z$ (resp. $\mathfrak {F}_w)$ in the wandering subspace $M \ominus zw{\mathfrak{M}}$ of $\mathfrak{M}$ with respect to the shift operator $zw$. If $\mathfrak{F}_w \ne \{0\}$ and $\mathfrak{F}_z \ne \{0\}$, then we consider the certain form of invariant subspaces $\mathfrak{M}$ of $L^2(T^2)$. Furthermore, we study certain classes of invariant subspaces of $L^2(T^2)$.