Frames of iterations and vector-valued model spaces

CABRELLI, C.; MOLTER, U.; SUAREZ, DANIEL

Sampling, Approximation, and Signal Analysis (Harmonic Analysis in the Spirit of J. Rowland Higgins)

Birkhauser Cham

Lugar: Washington; Año: 2023;

Let T be a bounded operator on a Hilbert space H, and F = { fj, j ∈ J} an at most countable set of vectors in H . In this note we characterize the pairs (T, F ) suchthat{Tnf : f ∈F,n∈I}formaframeofH,forthecasesofI=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 l2(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 L2(T, l2(J)).Furthermore, we characterize the frames of iterations for vector-valued model opera- tors 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. For the unilateral case we proved a formula for a lower bound. We conjecture that this lower bound is sharp. We give a new proof in the bilateral case, for which a formula is known.