The Isoperimetric Inequality for Partial Sums of Toeplitz Eigenvalues in the Fock Space

Abstract

Abstract. We prove that, among all subsets [Formula: see text] having circular symmetry and prescribed measure, the ball is the only maximizer of the sum of the first [Formula: see text] eigenvalues ([Formula: see text]) of the corresponding Toeplitz operator [Formula: see text] on the Fock space [Formula: see text]. As a byproduct, we prove that, again among circularly symmetric sets of prescribed measure, balls maximize any Schatten [Formula: see text]-norm of [Formula: see text] for [Formula: see text] (and minimize the corresponding quasinorm for [Formula: see text]), and that the second eigenvalue is maximized by a particular annulus. Moreover, we extend some of these results to general radial symbols in [Formula: see text] with [Formula: see text], characterizing those that maximize the sum of the first [Formula: see text] eigenvalues. We also show a symmetry breaking phenomenon for the second eigenvalue, when the assumption of circular symmetry is dropped.