Riesz Outer Product Hilbert Space Frames: Quantitative Bounds, Topological Properties, and Full Geometric Characterization

TL;DR

This paper provides a comprehensive analysis of outer product frames in Hilbert spaces, focusing on their bounds, independence, spanning properties, and geometric classification, with implications for frame theory and operator analysis.

Contribution

It offers the first detailed study of outer product frames induced by vector sequences, including bounds, independence, and a full geometric classification of dependent cases.

Findings

Riesz sequences of vectors produce Riesz sequences of outer products with comparable bounds.

Equiangular tight frames yield optimal Riesz bounds for outer products.

Almost all frames with certain bounds induce independent outer product sequences.

Abstract

Outer product frames are important objects in Hilbert space frame theory. But very little is known about them. In this paper, we make the first detailed study of the family of outer product frames induced directly by vector sequences. We are interested in both the quantitative attributs of these outer product sequences (in particular, their Riesz and frame bounds), as well as their independence and spanning properties. We show that Riesz sequences of vectors yield Riesz sequences of outer products with the same (or better) Riesz bounds. Equiangular tight frames are shown to produce Riesz sequences with optimal Riesz bounds for outer products. We provide constructions of frames which produce Riesz outer product bases with "good" Riesz bounds. We show that the family of unit norm frames which yield independent outer product sequences is open and dense (in a Euclidean-analytic sense) within the topological space $ \otimes_{i=1}^M S_{N - 1}$ where $M$ is less than or equal to the dimension of the space of symmetric operators on $\mathbb{H}^N$; that is to say, almost every frame with such a bound on its cardinality will induce a set of independent outer products. Thus, this would mean that finding the necessary and sufficient conditions such that the induced outer products are dependent is a more interesting question. For the coup de grace, we give a full analytic and geometric classification of such sequences which produce dependent outer products.