Parseval frames for ICC groups

TL;DR

This paper studies Parseval frames generated by ICC group actions on Hilbert spaces, providing parametrization, disjointness criteria, and undersampling results, with applications to subrepresentations of free groups.

Contribution

It introduces a parametrization of Parseval frames via operators in the commutant and characterizes disjointness, offering new insights into frames generated by ICC groups.

Findings

Parametrization of Parseval frames by commutant operators

Characterization of when two frames are strongly disjoint

Undersampling results linking frames to subgroup-generated orthonormal bases

Abstract

We analyze Parseval frames generated by the action of an ICC group on a Hilbert space. We parametrize the set of all such Parseval frames by operators in the commutant of the corresponding representation. We characterize when two such frames are strongly disjoint. We prove an undersampling result showing that if the representation has a Parseval frame of norm $\frac{1}{\sqrt{N}}$, the Hilbert space is spanned by an orthonormal basis generated by a subgroup. As applications we obtain some sufficient conditions under which a unitary representation admits a Parseval frame which is spanned by an Riesz sequences generated by a subgroup. In particular, every subrepresentation of the left regular representation of a free group has this property.