Certain invariant subspace structure of $L^{2}( \mathbb{T}^{2})$

In this note, we study certain structure of an invariant subspace $\mathfrak{M}$ of $L^{2}(\mathbb{T}^{2})$. Considering the largest $z$-invariant (resp. $w$-invariant) subspace in the wandering subspace $\mathfrak{M} \ominus zw \mathfrak{M}$ of $\mathfrak{M}$ with respect to the shift operator $zw$, we give an alternative characterization of Beurling-type invariant subspaces. Furthermore, we consider a certain class of invariant subspaces.