Frames from rescaled iterative system

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

   • Akram Aldroubi,
   • Carlos Cabrelli,
   • Ahmet F Cakmak,
   • Ursula Molter,
   • A. Petrosyan

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