Frames by orbits of two operators that commute

Frames formed by orbits of vectors through the iteration of a bounded operator have recently attracted considerable attention, in particular due to its applications to dynamical sampling. In this article, we consider two commuting bounded operators acting on some separable Hilbert space $\mathcal H$. We completely characterize operators $T$ and $L$ with $TL=LT$ and sets $\Phi\subset \mathcal H$ such that the collection $\{T^k L^j \phi: k\in \mathbb Z, j\in J, \phi \in \Phi \}$ forms a frame of $\mathcal H$. This is done in terms of model subspaces of the space of square integrable functions defined on the torus and having values in some Hardy space with multiplicity. The operators acting on these models are the bilateral shift and the compression of the unilateral shift (acting pointwisely). This context includes the case when the Hilbert space $\mathcal H$ is a subspace of $L^2(\mathbb R)$, invariant under translations along the integers, where the operator $T$ is the translation by one and $L$ is a shift-preserving operator.