Every Hilbert space frame has a Naimark complement

TL;DR

This paper demonstrates that all Hilbert space frames have Naimark complements with properties useful for various applications, correcting a previous mistake and expanding the scope beyond Parseval frames.

Contribution

It extends the concept of Naimark complements to all Hilbert space frames, not just Parseval frames, and corrects a related error in fusion frame literature.

Findings

All Hilbert space frames have Naimark complements with key properties.

Naimark complements can be used for equiangular frames, RIP, and fusion frames.

Corrects a previous mistake in computing chordal distances for Naimark complements.

Abstract

Naimark complements for Hilbert space Parseval frames are one of the most fundamental and useful results in the field of frame theory. We will show that actually all Hilbert space frames have Naimark complements which possess all the usual properties for Naimark complements with one notable exception. So these complements can be used for equiangular frames, RIP property, fusion frames etc. Along the way, we will correct a mistake in a recent fusion frame paper where chordal distances for Naimark complements are computed incorrectly.