Frames by Iterations and Invariant Subspaces

This paper presents a characterization of systems of iterations that generate frames of abstract separable Hilbert spaces. The characterization is achieved through a correspondence with a canonical system of iterations that form Parseval frames of certain subspaces of the space of vector-valued functions ${L^2}(\mathbb{T},\mathcal{K})$, where K is a Hardy space with multiplicity. These subspaces possess the property of being invariant under two shift operators with multiplicity.Furthermore, we provide a clear description of the subspaces generated by these canonical systems of iterations.