Frames of iterations and vector-valued model spaces

TL;DR

This paper characterizes when iterates of a bounded operator on a Hilbert space form a frame, linking it to model spaces, shift operators, and Toeplitz operators, and explores minimal orbit sets for frame generation.

Contribution

It provides a comprehensive characterization of frames generated by operator iterations, extending recent work to vector-valued model spaces and analyzing minimal orbit conditions.

Findings

Characterization of frames via shift and model space operators

Extension to vector-valued model operators with finite index sets

Analysis of minimal number of orbits needed for frame formation

Abstract

Let T be a bounded operator on a Hilbert space H, and F = {f_j: j in J} an at most countable set of vectors in H. In this note, we characterize the pairs {T, F} such that {T^n f: f in F, n in I} form a frame of H, for the cases of I = N_0 and I = Z. The characterization for unilateral iterations gives a similarity with the compression of the shift acting on model spaces of the Hardy space of analytic functions defined on the unit disk with values in $l^2(J). This generalizes recent work for iterations of a single function. In the case of bilateral iterations, the characterization is by the bilateral shift acting on doubly invariant subspaces of L^2(T,l^2(J)). Furthermore, we characterize the frames of iterations for vector-valued model operators when J is finite in terms of Toeplitz and multiplication operators in the unilateral and bilateral case, respectively. Finally, we study the problem of finding the minimal number of orbits that produce a frame in this context.