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Session S13 - Harmonic Analysis, Fractal Geometry, and Applications

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A model for frames of iterations of two operators

Alejandra Patricia Aguilera Aguilera

Universidad de Buenos Aires, IMAS-CONICET, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider frames in a separable Hilbert space $\mathcal{H}$ of the form $\{T^{k}L^{j}v: k\in\mathbb{Z}, j=0,1,2,...\}$ with a bounded operator $L$, a unitary operator $T$ that commutes with $L$, and a vector $v\in\mathcal{H}$.

We observe that $\mathcal{H}$ is mapped isomorphically to a subspace $\mathfrak{N}$ of the Hilbert space of measurable vector-valued functions $L^{2}(\mathbb{T},H^{2})$, where $H^{2}:=H^{2}(\mathbb{D})$ is the Hardy space on the unit disc. Then we show that the system $\{T^{k}L^{j}v: k\in\mathbb{Z}, j\in j=0,1,2,...\}$ corresponds to the set  $\{\mathcal{U}^{k} \hat{A}^{j}\psi: k\in\mathbb{Z}, j\in j=0,1,2,...\}$ via the underlying isomorphism between $\mathcal{H}$ and $\mathfrak{N}$. Here, $\psi$ is some function in $\mathfrak{N}$, $\mathcal{U}$ is the bilateral shift with multiplicity in $L^{2}(\mathbb{T},H^{2})$, and $\hat{A}$ commutes with $\mathcal{U}$ and acts pointwise as the compression of the unilateral shift on model subspaces of the Hardy space $H^2$.

We also give a characterization of all vectors $v$ such that the system $\{T^{k}L^{j}v: k\in\mathbb{Z}, j=0,1,2,...\}$ is a frame for $\mathcal{H}$ assuming that there is some $v_{0}\in\mathcal{H}$ with that property.

Joint work with Carlos Cabrelli (Universidad de Buenos Aires, IMAS-CONICET), Diana Carbajal (Universidad de Buenos Aires, IMAS-CONICET) and Victoria Paternostro (Universidad de Buenos Aires, IMAS-CONICET).

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