The isoperimetric inequality for partial sums of Toeplitz eigenvalues in the Fock space
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| Format: | Preprint |
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2025
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| _version_ | 1866912440952815616 |
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| author | Nicola, Fabio Riccardi, Federico Tilli, Paolo |
| author_facet | Nicola, Fabio Riccardi, Federico Tilli, Paolo |
| contents | We prove that, among all subsets $Ω\subset \mathbb{C}$ 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 $T_Ω$ on the Fock space $\mathcal{F}$. As a byproduct, we prove that balls maximize any Schatten $p$-norm of $T_Ω$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07069 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The isoperimetric inequality for partial sums of Toeplitz eigenvalues in the Fock space Nicola, Fabio Riccardi, Federico Tilli, Paolo Functional Analysis Classical Analysis and ODEs We prove that, among all subsets $Ω\subset \mathbb{C}$ 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 $T_Ω$ on the Fock space $\mathcal{F}$. As a byproduct, we prove that balls maximize any Schatten $p$-norm of $T_Ω$ 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. |
| title | The isoperimetric inequality for partial sums of Toeplitz eigenvalues in the Fock space |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2503.07069 |