Frames by orbits of two operators that commute

AGUILERA, A.; CABRELLI, C.; CARBAJAL, D.; PATERNOSTRO, V.

APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2023 vol. 66 p. 46 - 61

1063-5203

Frames formed by orbits of vectors through the iteration of a bounded operator have recently attracted considerable attention, particularly due to its applications to dynamical sampling. This article considers two commuting bounded operators acting on some separable Hilbert space H. We characterize entirely operators T and L with TL = LT and sets Φ ⊂ H such that the collection {TkLjφ : k ∈ Z,j ∈ J,φ ∈ Φ} forms a frame of H. This is done in terms of model subspaces of the space of square-integrable functions defined on the torus and having values in some Hardy space with multiplicity. The operators acting on these models are the bilateral shift and the compression of the unilateral shift (acting point wisely). This context includes the case when the Hilbert space H is a subspace of L2(R), invariant under translations along the integers, where the operator T is the translation by one and L is a shift-preserving operator.