The normalized orbit of a bounded normal operator can be a frame

Abstract

Conjecture 3 in [A. Aldroubi, C. Cabrelli, I. Krishtal, and U. Molter, Dynamical Sampling: A Survey, La Matematica 5 (2026), Article 37] postulates that for any bounded normal operator $T$ on a Hilbert space $H$ and any vector $g\in H$ the system \[ \left\{\frac{T^k g}{\|T^k g\|}: k=0,1,2,\ldots\right\} \] is not a frame. It was motivated by [A. Aldroubi, C. Cabrelli, A. F. Çakmak, U. Molter, and A. Petrosyan, Iterative actions of normal operators, J. Funct. Anal. 272 (2017), no. 3, 1121--1146], where it was established that such frames do not exist when $T$ is a self adjoint operator. We show, however, that this conjecture is false by presenting a construction of $H$, $T$, and $g$ such that the normalized orbit considered is indeed a frame. The operator is diagonal and is defined via a decomposition of the space into finite blocks rapidly increasing in size. We also provide an $ε$-perturbation $S$ of the operator $T$ such that the system \[ \left\{{S^k g}: k=0,1,2,\ldots\right\} \] is a Carleson frame in the sense of [A. Aldroubi, C. Cabrelli, U. Molter, and S. Tang, Dynamical sampling, Appl. Comput. Harmon. Anal. 42 (2017), no. 3, 378--401] and [O. Christensen, M. Hasannasab, F. M. Philipp, and D. Stoeva, The mystery of Carleson frames, Appl. Comput. Harmon. Anal. 72 (2024), Article 101659]. The constructions were achieved using ChatGPT, whose assistance was also employed in the preparation of this manuscript.