Toeplitz Quantization and Convexity

Abstract

Let $T_f^m$ be the Toeplitz quantization of a real $C^{\infty }$ function defined on the sphere $\mathbb{CP}(1)$. $T_f^m$ is therefore a Hermitian matrix with spectrum ${\lambda }^m=({\lambda }_0^m,\dots ,{\lambda }_m^m)$. Schur's theorem says that the diagonal of a Hermitian matrix $A$ that has the same spectrum of $T_f^m$ lies inside a finite dimensional convex set whose extreme points are $\{({\lambda }_{\sigma (0)}^m,\dots ,{\lambda }_{\sigma (m)}^m)\}$, 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 $f^{\ast }\circ \varphi$, where $f^{\ast }$ is the decreasing rearrangement of $f$ and $\varphi$ ranges over the set of measure preserving transformations of the unit interval $[0,1]$.