Dynamical sampling and frame representations with bounded operators

The purpose of this paper is to study frames for a Hilbert space ${\cal H},$ having the form $\{T^n \varphi\}_{n=0}^\infty$ for some $\varphi \in {\cal H}$ and an operator $T: {\cal H} \to {\cal H}.$ We characterize the frames that have such a representation for a bounded operator $T,$ and discuss the properties of this operator. In particular, we prove that the image chain of $T$ has finite length $N$ in the overcomplete case; furthermore $\{T^n \varphi\}_{n=0}^\infty$ has the very particular property that $\{T^n \varphi\}_{n=0}^{N-1} \cup \{T^n \varphi\}_{n=N+\ell}^\infty$ is a frame for ${\cal H}$ for all $\ell\in {\mathbf N}_0$. We also prove that frames of the form $\{T^n \varphi\}_{n=0}^\infty$ are sensitive to the ordering of the elements and to norm-perturbations of the generator $\varphi$ and the operator $T.$ On the other hand positive stability results are obtained by considering perturbations of the generator $\varphi$ belonging to an invariant subspace on which $T$ is a contraction.