Dynamical sampling and frame representations with bounded operators

The purpose of this paper is to study frames for a Hilbert space H, having the form {Tnφ}n=0∞ for some φ∈H and an operator T:H→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 {Tnφ}n=0∞ has the very particular property that {Tnφ}n=0N−1∪{Tnφ}n=N+ℓ∞ is a frame for H for all ℓ∈N0. We also prove that frames of the form {Tnφ}n=0∞ are sensitive to the ordering of the elements and to norm-perturbations of the generator φ and the operator T. On the other hand positive stability results are obtained by considering perturbations of the generator φ belonging to an invariant subspace on which T is a contraction.