Frames via Operator Orbits: Structure of Subspaces Determining Boundedness

Motivated by recent work related to Dynamical Sampling by Christensen and Hasannasab, 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 $T \in B(H)$. In particular, we show that a frame, $\{f_n\}_{n \in \mathbb{N}_0}$, admits the form $\{T^{n} \varphi \}_{n \geq 0}$ with $T \in B(H)$ if and only if the frame is a Riesz basis or it is linearly independent and the system, $\{S^nc\}_{n \in \mathbb{N}_{0}}$, is a Parseval frame for the kernel of the synthesis operator (where $S$ is the right-shift operator on $\ell^2(\mathbb{N}_{0})$ and $c$ is some fixed scalar sequence)