A counterexample to the Berger--Coburn conjecture
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918311910965248 |
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| author | Looi, Sam |
| author_facet | Looi, Sam |
| contents | Berger and Coburn proposed an endpoint boundedness criterion for Toeplitz operators on the Bargmann--Fock space in which the decisive quantity is the heat transform of the symbol at the borderline time $t=\tfrac14$, the time naturally singled out by the Weyl calculus under the Bargmann transform. We show that this criterion fails for general measurable symbols in every complex dimension $n\ge 1$. Concretely, we construct a measurable symbol $g\in L^2(\mathbb C^n,dμ)$ such that $gk_a\in L^2(dμ)$ for every normalized reproducing kernel $k_a$, and the associated Toeplitz form extends to a bounded operator on $H^2(\mathbb C^n,dμ)$, but the heat transform $g^{(1/4)}$ is unbounded on $\mathbb C^n$. The example is obtained by summing translated bounded "blocks" whose Toeplitz norms are summable while their $t=\tfrac14$ heat profiles have fixed size. The blocks are produced by combining a Hilbert--Schmidt estimate for Weyl quantization with the Bargmann correspondence between Weyl and Toeplitz operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20859 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A counterexample to the Berger--Coburn conjecture Looi, Sam Functional Analysis Analysis of PDEs Complex Variables Spectral Theory Berger and Coburn proposed an endpoint boundedness criterion for Toeplitz operators on the Bargmann--Fock space in which the decisive quantity is the heat transform of the symbol at the borderline time $t=\tfrac14$, the time naturally singled out by the Weyl calculus under the Bargmann transform. We show that this criterion fails for general measurable symbols in every complex dimension $n\ge 1$. Concretely, we construct a measurable symbol $g\in L^2(\mathbb C^n,dμ)$ such that $gk_a\in L^2(dμ)$ for every normalized reproducing kernel $k_a$, and the associated Toeplitz form extends to a bounded operator on $H^2(\mathbb C^n,dμ)$, but the heat transform $g^{(1/4)}$ is unbounded on $\mathbb C^n$. The example is obtained by summing translated bounded "blocks" whose Toeplitz norms are summable while their $t=\tfrac14$ heat profiles have fixed size. The blocks are produced by combining a Hilbert--Schmidt estimate for Weyl quantization with the Bargmann correspondence between Weyl and Toeplitz operators. |
| title | A counterexample to the Berger--Coburn conjecture |
| topic | Functional Analysis Analysis of PDEs Complex Variables Spectral Theory |
| url | https://arxiv.org/abs/2601.20859 |