The isoperimetric inequality for partial sums of Toeplitz eigenvalues in the Fock space

2025/03/10 by Nicola, Fabio, Riccardi, Federico, Tilli, Paolo · 1 citation
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2503.07069

Abstract

We prove that, among all subsets Ω⊂ ℂ having circular symmetry and prescribed measure, the ball is the only maximizer of the sum of the first K eigenvalues (K≥ 1) of the corresponding Toeplitz operator T_Ω on the Fock space F. As a byproduct, we prove that balls maximize any Schatten p-norm of T_Ω for p>1 (and minimize the corresponding quasinorm for p<1), and that the second eigenvalue is maximized by a particular annulus. Moreover, we extend some of these results to general radial symbols in L^p(ℂ), with p > 1, characterizing those that maximize the sum of the first K eigenvalues. We also show a symmetry breaking phenomenon for the second eigenvalue, when the assumption of circular symmetry is dropped.

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