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

We prove that, among all subsets $\mathrm{\Omega }\subset \mathbb{C}$ having circular symmetry and prescribed measure, the ball is the only maximizer of the sum of the first $K$ eigenvalues ($K\ge 1$) of the corresponding Toeplitz operator $T_{\mathrm{\Omega }}$ on the Fock space $\mathcal{F}$. As a byproduct, we prove that balls maximize any Schatten $p$-norm of $T_{\mathrm{\Omega }}$ 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.