Submodules of $H^2(\mathbb{T}^2)$ and Frames by Pairs of Bounded Commuting Operators

TL;DR

This paper characterizes when pairs of bounded commuting operators generate frames via unilateral iterations on a single vector, linking the problem to submodules of the Hardy space and two-variable Jordan blocks.

Contribution

It provides a necessary and sufficient condition for such frames, connecting operator theory with the structure of submodules in Hardy spaces.

Findings

Characterization of frames generated by pairs of commuting operators

Connection between frames and submodules of $H^2(\

Abstract

Recent work in Dynamical Sampling has been centered on characterizing frames obtained by the orbit of a vector under a bounded operator. We prove a necessary and sufficient condition for a pair of bounded commuting operators on a separable infinite-dimensional Hilbert space to generate a frame by unilateral iterations on a single vector. Applying the theory on submodules of the Hardy module $H^2(\mathbb{T}^2)$, we characterize these frames in terms of their relation to the two-variable Jordan block on a certain quotient module and provide some properties of frames of this form.