AI 中文总结
该研究证明面积固定的可测集中圆盘使Fock空间上Toeplitz算子的Ky Fan范数最大，结合Ky Fan极大值原理与量子信息方法完成证明，还得到Schatten和的等周不等式，验证了相关猜想。
AI 中文摘要
我们证明，在所有面积为有限值$s$的可测集$\Omega\subset \mathbb{C}$中，面积为$s$的圆盘能使Ky Fan范数达到最大；该范数定义为Fock空间上以$\mathbf{1}_{\Omega}$为符号的Toeplitz算子的前$N$个特征值之和。当$N=1$时，该结果退化为Nicola-Tilli著名的Faber--Krahn不等式；对于一般的$N$，此结论曾由Nicola、Riccardi和Tilli提出猜想，且他们已针对径向集证明了该猜想。我们的证明将Ky Fan极大值原理与基于密度算子Fock重排的量子信息论方法相结合。作为副产品，我们得到了$0<p\leq \infty$范围内Schatten和的等周不等式。
英文摘要
We show that, among all measurable sets $Ω\subset \mathbb{C}$ with finite area $s$, the disc of area $s$ maximizes the Ky Fan norm, which is defined as the sum of the first $N$ eigenvalues of the Toeplitz operator with symbol $\mathbf{1}_{Ω}$ on the Fock space. For $N=1$ this reduces to Nicola-Tilli's celebrated Faber--Krahn inequality and for general $N$ the result was conjectured by Nicola, Riccardi and Tilli, who proved it for radial sets. The proof combines Ky Fan's maximum principle with methods from quantum information theory based on Fock rearrangements of density operators. As a by-product, we obtain isoperimetric inequalities for the Schatten sums in the range $0<p\leq \infty $.
