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arXiv · 2610.19545

Normalized Orbit Frames and Essential Spectrum, II

James Tian
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Abstract

Given $0<r_{j}<1$ with $r_{j}\to1$, there exist a normal operator $T$ on $\ell^{2}$ with eigenvalues $λ_{j}$ satisfying $|λ_{j}|=r_{j}$ and $g\in\ell^{2}$ such that \[ \left\{ \frac{T^{n}g}{\left\Vert T^{n}g\right\Vert }\right\} _{n\geq0}\text{is a frame}\:\Longleftrightarrow\:\sum_{j\geq1}\left(1-r^{2}_{j}\right)=\infty. \] Whenever this condition holds, the frame bounds can be chosen with ratio arbitrarily close to one.

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James Tian. 2026-10-14. Normalized Orbit Frames and Essential Spectrum, II. https://arxiv.org/abs/2610.19545

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