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<p>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 <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1822_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^2(\mathbb {T}^2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, 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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年9月24日

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  • Victor Bailey,
  • Carlos Cabrelli

摘要

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 \(H^2(\mathbb {T}^2)\) H 2 ( 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.