Coarse comparison of Bessel families

Abstract

A Bessel family is a collection of vectors of a Hilbert space whose coefficients are square summable; orthonormal bases and frames are examples. We compare two such families by asking when the size of the coefficients of one is controlled, up to constants, by that of the other after a change of variables taken from a fixed class $\mathcal{O}$ of operators. Each class (bounded, injective, bounded below, invertible, Fredholm, and others) gives an equivalence relation that ignores redundancy and treats frames and nonframes on the same footing. For every relation we determine the classes by explicit spectral data: the singular numbers of the analysis operator, its kernel, the closedness of its range, and the multiplicities of its spectrum at each dyadic scale. We also characterize the families equivalent to an orthonormal basis, compute an operator Banach--Mazur distance, and prove a Paley--Wiener stability result that needs no frame assumption. These relations are instances of the comparison categories of a companion article.

arXiv commentAI-generated from an initial draft and supervision by the author. 30 pages