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

Abstract:

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.