Group Riesz and Frame Sequences: The Bracket and the Gramian

TL;DR

This paper demonstrates that the operator Bracket map and Gramian are equivalent on a dense set in Hilbert spaces for group representations, simplifying the characterization of Riesz and frame sequences.

Contribution

It establishes the equivalence of Bracket map and Gramian in the context of group representations, unifying previous characterizations of Riesz and frame sequences.

Findings

Bracket map and Gramian coincide on a dense set

Simplifies characterization of Riesz and frame sequences

Unifies previous results in frame theory

Abstract

Given a discrete group and a unitary representation on a Hilbert space $\mathcal{H}$, we prove that the notions of operator Bracket map and Gramian coincide on a dense set of $\mathcal{H}$. As a consequence, combining this result with known frame theory, we can recover in a simple way all previous Bracket characterizations of Riesz and frame sequences generated by a single element under a unitary representation.