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arXiv · 2507.04992

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

Victor Bailey·Carlos Cabrelli
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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.

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Victor Bailey, Carlos Cabrelli. 2025-07-07. Submodules of $H^2(\mathbb{T}^2)$ and Frames by Pairs of Bounded Commuting Operators. https://arxiv.org/abs/2507.04992

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