The paper The Faber-Krahn inequality for partial sums of eigenvalues of Toeplitz operators, by Fabio Nicola, Federico Riccardi, and Paolo Tilli, is now available on arXiv (arXiv:2503.07069).
We prove that, among all radial subsets \(\Omega\subset \mathbb{C}\) of 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 \(\mathcal{F}^2(\mathbb{C})\). 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.