Spanning and independence properties of frame partitions

TL;DR

This paper investigates how frames in Hilbert spaces can be partitioned into sets with properties like independence and spanning, providing new results for both finite and infinite dimensions under certain conditions.

Contribution

It establishes new decomposition results for Parseval frames, including independence and spanning properties, with implications for finite and infinite dimensional spaces, assuming the Kadison-Singer conjecture.

Findings

Frames can be decomposed into sets with spanning complements in finite dimensions.

Under the Kadison-Singer conjecture, similar decompositions hold in infinite dimensions.

Small-norm Parseval frames allow for partitions with independent sets and multiple spanning subsets.

Abstract

We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that in finite dimensional Hilbert spaces, Parseval frames with norms bounded away from 1 can be decomposed into a number of sets whose complements are spanning, where the number of these sets only depends on the norm bound. We also prove, assuming the Kadison-Singer conjecture is true, that this holds for infinite dimensional Hilbert spaces. Further, we prove a stronger result for Parseval frames whose norms are uniformly small, which shows that in addition to the spanning property, the sets can be chosen to be independent, and the complement of each set to contain a number of disjoint, spanning sets.