Frames of subspaces and operators

TL;DR

This paper explores the relationships between operators, orthonormal bases of subspaces, and fusion frames in Hilbert spaces, providing new conditions, generalizations, and insights into the structure and weights of such frames.

Contribution

It introduces new sufficient conditions for constructing frames of subspaces via operators and orthonormal bases, generalizes previous results on oblique projections, and studies the admissible weights for generating sequences.

Findings

Provided conditions for frames of subspaces using operators and bases

Generalized results relating frames of subspaces and oblique projections

Analyzed the set of admissible weights for generating subspace sequences

Abstract

We study the relationship between operators, orthonormal basis of subspaces and frames of subspaces (also called fusion frames) for a separable Hilbert space $\mathcal{H}$. We get sufficient conditions on an orthonormal basis of subspaces $\mathcal{E} = \{E_i \}_{i\in I}$ of a Hilbert space $\mathcal{K}$ and a surjective $T\in L(\mathcal{K}, \mathcal{H})$ in order that $\{T(E_i)\}_{i\in I}$ is a frame of subspaces with respect to a computable sequence of weights. We also obtain generalizations of results in [J. A. Antezana, G. Corach, M. Ruiz and D. Stojanoff, Oblique projections and frames. Proc. Amer. Math. Soc. 134 (2006), 1031-1037], which related frames of subspaces (including the computation of their weights) and oblique projections. The notion of refinament of a fusion frame is defined and used to obtain results about the excess of such frames. We study the set of admissible weights for a generating sequence of subspaces. Several examples are given.