Frame Properties of Operator Orbits

Abstract: We consider sequences in a Hilbert space $\mathcal H$ of the form $(T^nf_0)_{n\in I},$ with a linear operator $T$, the index set being either $I = \mathbb N$ or $I = \mathbb Z$, a vector $f_0\in \mathcal H$, and answer the following two related questions: (a) {\it Which frames for $\mathcal H$ are of this form with an at least closable operator $T$?} and (b) {\it For which bounded operators $T$ and vectors $f_0$ is $(T^nf_0)_{n\in I}$ a frame for $\mathcal H$?} As a consequence of our results, it turns out that an overcomplete Gabor or wavelet frame can never be written in the form $(T^nf_0)_{n\in\mathbb N}$ with a bounded operator $T$. The corresponding problem for $I = \mathbb Z$ remains open. Despite the negative result for Gabor and wavelet frames, the results demonstrate that the class of frames that can be represented in the form $(T^nf_0)_{n\in\mathbb N}$ with a bounded operator $T$ is significantly larger than what could be expected from the examples known so far.