Toeplitz Quantization and Convexity

TL;DR

This paper studies the asymptotic behavior of convex sets associated with Toeplitz quantizations of smooth functions on the sphere, revealing their convergence to a large convex set characterized by measure-preserving transformations.

Contribution

It proves that the convex sets formed by the spectra of Toeplitz matrices converge to a convex set in L^2([0,1]) with extreme points given by rearrangements of the original function.

Findings

Convex sets from Toeplitz spectra converge in L^2 to a large convex set.

Extreme points of the limit set are functions composed with measure-preserving transformations.

The result links spectral properties of Toeplitz matrices to measure theory and rearrangements.

Abstract

Let $T^m_f $ be the Toeplitz quantization of a real $ C^{\infty}$ function defined on the sphere $ \mathbb{CP}(1)$. $T^m_f $ is therefore a Hermitian matrix with spectrum $\lambda^m= (\lambda_0^m,\ldots,\lambda_m^m)$. Schur's theorem says that the diagonal of a Hermitian matrix $A$ that has the same spectrum of $ T^m_f $ lies inside a finite dimensional convex set whose extreme points are $\{( \lambda_{\sigma(0)}^m,\ldots,\lambda_{\sigma(m)}^m)\}$, where $\sigma$ is any permutation of $(m+1)$ elements. In this paper, we prove that these convex sets "converge" to a huge convex set in $L^2([0,1])$ whose extreme points are $ f^*\circ \phi$, where $ f^*$ is the decreasing rearrangement of $ f$ and $ \phi $ ranges over the set of measure preserving transformations of the unit interval $ [0,1]$.