Frame potential for finite-dimensional Banach spaces

TL;DR

This paper introduces a new concept called the frame potential for finite-dimensional Banach spaces, characterizes finite unit norm tight frames (FUNTFs), and proves their existence in various Banach spaces, extending Hilbert space results.

Contribution

It defines the frame potential for Banach spaces, characterizes FUNTFs in this setting, and proves their existence in spaces with unconditional bases, advancing the understanding of frame theory beyond Hilbert spaces.

Findings

Frame potential characterizes FUNTFs in Banach spaces.

Existence of FUNTFs in spaces with 1-unconditional bases.

Open questions remain on FUNTFs in Banach spaces.

Abstract

We define the frame potential for a Schauder frame on a finite dimensional Banach space as the square of the $2$-summing norm of the frame operator. As is the case for frames for Hilbert spaces, we prove that the frame potential can be used to characterize finite unit norm tight frames (FUNTFs) for finite dimensional Banach spaces. We prove the existence of FUNTFs for a variety of spaces, and in particular that every $n$-dimensional complex Banach space with a $1$-unconditional basis has a FUNTF of $N$ vectors for every $N\geq n$. However, many interesting results on FUNTFs and sums of rank one projections for Hilbert spaces remain unknown for Banach spaces and we conclude the paper with multiple open questions.