Existence and Properties of Frames of Iterations

Abstract: Let T be a bounded operator acting on an infinite-dimensional Hilbert space. We provide necessary and sufficient conditions for the existence of a Parseval frame of iterations generated by T. Based in this result, we give a new proof of a known characterization concerning the existence of (not necessarily Parseval) frames of iterations. As a consequence of our results, we obtain that certain operators lack Parseval frames despite possessing frames of iterations. In addition, we introduce the index of a bounded operator T as the minimal number of vectors required to generate a frame via its iterations. We obtain the exact value of this index for both Parseval frames and general frames of iterations, and we give a constructive method for generating such frames. Assuming that T satisfies the conditions ensuring that both T and T^* admit frames of iterations, we show how to construct Parseval frames generated by iterations of T and also by iterations of T^*. This construction relies on universal models in vector-valued Hardy spaces and the theory of universal dilations. Furthermore, we provide necessary and sufficient conditions under which the frames generated by T and T^* are similar, in terms of the inner function of the associated model in the Hardy space of the operator T.