Fusion frame constructions and frame partitions

Abstract

Fusion frames consist of a sequence of subspaces from a Hilbert space and corresponding positive weights so that the sum of weighted orthogonal projections onto these subspaces is an invertible operator on the space. Despite extensive literature on fusion frames, and several construction methods for unit-weight fusion frames with prescribed subspace dimensions and fusion frame operator spectra, there do not exist such constructions for prescribed non-unit weights. There are also very few constructions which allow one to control geometric properties among the subspaces. First we will adapt a flexible construction technique known as spectral tetris to provide the first constructions of the most general classes of fusion frames: fusion frames with arbitrary non-unit weights. Moreover, we provide for the first time necessary and sufficient conditions for when a fusion frame can be constructed via spectral tetris methods. Then we present a new alternative construction leveraging Naimark complements to build a large class of fusion frames whose principal angles between any two subspaces are constant. Changing focus to frame partitions, the celebrated Rado-Horn theorem is an often-applied tool which provides a tight bound on the minimal number of linearly indpendent sets in a partition of vectors. This theorem has been rediscovered many times over, but no existing results describe how to find such partitions. We present an alternative proof of the Rado-Horn theorem and then adapt our proof's ideas to capture much more spanning and independence information compared with existing Rado-Horn results. We further describe how to build partitions with these properties.