Bessel orbits of normal operators

Friedrich Philipp

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arXiv · 1605.07299

Bessel orbits of normal operators

Friedrich Philipp

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Abstract

Given a bounded normal operator $A$ in a Hilbert space and a fixed vector $x$, we elaborate on the problem of finding necessary and sufficient conditions under which $(A^kx)_{k\in\mathbb N}$ constitutes a Bessel sequence. We provide a characterization in terms of the measure $\|E(\cdot)x\|^2$, where $E$ is the spectral measure of the operator $A$. In the separately treated special cases where $A$ is unitary or selfadjoint we obtain more explicit characterizations. Finally, we apply our results to a sequence $(A^kx)_{k\in\mathbb N}$, where $A$ arises from the heat equation. The problem is motivated by and related to the new field of Dynamical Sampling which was recently initiated by Aldroubi et al.

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BibTeX RIS

Friedrich Philipp. 2016-05-24. Bessel orbits of normal operators. https://arxiv.org/abs/1605.07299

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