Continuous and discrete dynamical sampling

In this paper we study the continuous dynamical sampling problem at infinite time in a complex Hilbert space H. We find necessary and sufficient conditions on a bounded linear operator A∈B(H) and a set of vectors G⊂H, in order to obtain that {etAg}g∈G,t∈[0,∞) is a semi-continuous frame for H. We study if it is possible to discretize the time variable t and still have a frame for H. We also relate the continuous iteration etA on a set G to the discrete iteration (A′)n on G′ for an adequate operator A′ and set G′⊂H.