Frames via Unilateral Iterations of a Bounded Operator

Motivated by recent work in Dynamical Sampling, we prove a necessary and sufficient condition for a frame in a separable and infinite-dimensional Hilbert space to admit the form $\{T^{n} \varphi \}_{n \geq 0}$ with $T \in B(H)$. Also, a characterization of all the vectors $\varphi$ for which $\{T^{n} \varphi \}_{n \geq 0}$ is a frame for some $T \in B(H)$ is provided. Some auxiliary results on operator representations of Riesz frames are given as well.