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

Abstract

We prove that, among all subsets $\Omega\subset \mathbb{C}$ having circular symmetry and prescribed measure, the ball is the only maximizer of the sum of the first $K$ eigenvalues ($K\geq 1$) of the corresponding Toeplitz operator $T_\Omega$ on the Fock space $\mathcal{F}$. As a byproduct, we prove that 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 maximized by a particular annulus. Moreover, we extend some of these results to general radial symbols in $L^p(\mathbb{C})$, 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.