Commutators of spectral projections of spin operators

TL;DR

This paper proves that the operator norm of commutators of spectral projections of spin operators approaches 1/2 in the semiclassical limit, using Hankel operator theory and discussing related finite group cases.

Contribution

It introduces a novel proof for the asymptotic behavior of spectral projection commutators of spin operators using Hardy space Hankel operators.

Findings

Operator norm converges to 1/2 in the semiclassical limit

Hankel operators on Hardy space are instrumental in the proof

Includes discussion of analogous results for finite Heisenberg groups

Abstract

We present a proof that the operator norm of the commutator of certain spectral projections associated with spin operators converges to $\frac 1 2$ in the semiclassical limit. The ranges of the projections are spanned by all eigenvectors corresponding to positive eigenvalues. The proof involves the theory of Hankel operators on the Hardy space. A discussion of several analogous results is also included, with an emphasis on the case of finite Heisenberg groups.