Spectrum of normal operators that generate certain scalable iterative systems

Abstract

Let $A:H\to H$ be a normal operator on an infinite-dimensional separable Hilbert space $H$ and let $S\subseteq H$ be a finite subset such that $\{A^{n}x{\}}_{n\ge 0,x\in S}$ can be rescaled to form a frame for $H$. That is, there exist some subsets $J_x\subseteq \mathbb{N}\cup \{0\}$ and some set of nonzero scalars $(c_{n,x}{)}_{n\in J_x,x\in S}$ such that $\{c_{n,x}{A}^{n}x{\}}_{n\in J_x,x\in S}$ forms a frame for $H.$ Assume that there exist some $\hat{I}\cdot \in \mathbb{N}$ and $\hat{I}´>0$ such that for each infinite $J_x$ there is an increasing syndetic subsequence $(n_k^x{)}_{k\in \mathbb{N}}\subseteq J_x$ satisfying $|c_{n_k^x,x}|\Vert A^{i_k^x}x\Vert \ge \hat{I}´$ for some non-negative integers $i_k^x$ with $|i_k^x-n_k^x|\le \hat{I}\cdot$ for all $k\in \mathbb{N}$. We prove that there exist finitely many numbers $(r_i{)}_{i=1}^N$ such that the continuous spectrum of $A$ is concentrated on arcs of a circle centered at origin with radius $r_i$. In particular, $A$ must be a diagonal operator if $S$ is a singleton. As an application, we establish the conjecture proposed by Aldroubi et al.\ asserting that the iterative system $\{\frac{A^{n}x}{\Vert A^{n}x\Vert }{\}}_{n\ge 0,x\in S}$ is never a frame for $H$, provided one of the following two conditions holds: (i) The continuous spectrum of $A$ contains more than $|S|-1$ points with distinct moduli; (ii) $S$ is a singleton and $A$ is not a diagonal operator

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