Bracket map for Heisenberg group and the characterization of cyclic subspaces

TL;DR

This paper extends the concept of bracket maps from abelian groups to the noncommutative Heisenberg group, characterizing bases and frames for cyclic subspaces and suggesting generalizations to other non-abelian nilpotent Lie groups.

Contribution

It introduces a noncommutative bracket map framework for the Heisenberg group and characterizes cyclic subspace bases and frames, extending previous abelian results.

Findings

Bracket maps for the Heisenberg group are characterized.

Criteria for bases and frames in cyclic subspaces are established.

Results are generalized to certain non-abelian nilpotent Lie groups.

Abstract

The bracket map was originally considered for locally compact abelian groups. In this work we extend the study of bracket maps to the noncommutative setting, providing characterizations of bases and frames for cyclic subspaces of the Heisenberg group. We also indicate how to generalize these results to a class of non-abelian nilpotent Lie groups whose irreducible representations are square integrable modulo the center.