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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 thathave such a representation for a bounded operator $T,$ and discuss theproperties 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 theform $\{T^n \varphi\}_{n=0}^\infty$ are sensitive to the ordering of theelements and to norm-perturbations of the generator $\varphi$ and the operator$T.$ On the other hand positive stability results are obtained by consideringperturbations of the generator $\varphi$ belonging to an invariant subspace onwhich $T$ is a contraction.

Dynamical sampling and frame representations with bounded operators