Dynamical sampling: mixed frame operators, representations and perturbations

Motivated by recent progress in operator representation of frames, we investigate the frames of the form $\{T^n\phi {\}}_{n\in I}$ for $I=\mathbb{N},\mathbb{Z}$, and answer questions about representations, perturbations and frames induced by the action of powers of bounded linear operators. As a particular case, we discuss problems concerning representation of frames in terms of iterations of the mixed frame operators. As our another contribution, we consider frames of the form $ {a_n T^n \varphi}{n=0}^{\infty} $forsomenon-zeroscalars$ {a_n}{n=0}^{\infty} $, and we obtain some new results in dynamical sampling. Finally, we will present some auxiliary results related to the perturbation of sequences of the form $\{T^n\phi {\}}_{n=0}^{\mathrm{\infty }}$.