# Frames from rescaled iterative system

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Posted online: 2018-12-31 01:50:36Z by Armenak Petrosyan25

Cite as: P-181231.1

• Functional Analysis

### Problem's Description

Let $A$ be a bounded linear operator on Hilbert space $\mathcal{H}$. For some fixed $x\in \mathcal{H}$, consider a system of vectors $$\left\{\frac{\|A^n x\|}{\|x\|} \right\}_{n=0,1,2,\dots}.$$ There are bounded (not-normal) operators for which a system of above form is an orthonormal basis (e.g. shift operator on $\mathcal{l}(\mathbb{N})$).

For self-adjoint operators it is known that this system cannot be a frame for any $\mathcal{H}$. Is the same true for any normal operator $A$?

1. ## Article Iterative actions of normal operators

Journal of Functional Analysis 272 (3), 1121-1146, 2017