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

We prove that, among all subsets \Omega \subset \BbbC having circular symmetry and prescribedmeasure, the ball is the only maximizer of the sum of the first \mathrm{K} eigenvalues (\mathrm{K} \geq 1) of the corre-sponding Toeplitz operator T\Omega on the Fock space \scrF . As a byproduct, we prove that, again amongcircularly symmetric sets of prescribed measure, balls maximize any Schatten p-norm of T\Omega for p > 1(and minimize the corresponding quasinorm for p < 1), and that the second eigenvalue is maximizedby a particular annulus. Moreover, we extend some of these results to general radial symbols inLp(\BbbC ) with p > 1, characterizing those that maximize the sum of the first \mathrm{K} eigenvalues. We alsoshow a symmetry breaking phenomenon for the second eigenvalue, when the assumption of circularsymmetry is dropped