Toeplitz Quantization and Convexity

Original · EN

Let Tᵐf be the Toeplitz quantization of a real C∞ function defined on the sphere CP(1). Tᵐf is therefore a Hermitian matrix with spectrum λᵐ= (λ₀ᵐ,,λₘᵐ). Schur's theorem says that the diagonal of a Hermitian matrix A that has the same spectrum of Tᵐf lies inside a finite dimensional convex set whose extreme points are {(λσ₍₀₎ᵐ,,λσ₍ₘ₎ᵐ)}, where σ is any permutation of (m+1) elements. In this paper, we prove that these convex sets "converge" to a huge convex set in L²([0,1]) whose extreme points are f*∘ ϕ, where f* is the decreasing rearrangement of f and ϕ ranges over the set of measure preserving transformations of the unit interval [0,1].