Zero-product problem for Toeplitz operators on the Fock space

@misc{qin2026zeroproduct,
  title = {{Zero-product problem for Toeplitz operators on the Fock space}},
  author = {Jie Qin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20555},
  eprint = {2609.20555},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20555},
}

TY  - UNPB
TI  - Zero-product problem for Toeplitz operators on the Fock space
AU  - Jie Qin
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20555
UR  - https://arxiv.org/abs/2609.20555
AB  - We answer Bauer and Le's question on zero products of Toeplitz operators on the Fock space F^2(ℂ^n)[JFA, 261 (2011), 9, 2617--2640]. For n≥2, we construct two bounded nonradial Schwartz symbols on ℂ^n whose Toeplitz operators are nonzero and have zero product on F^2(ℂ^n). For n=1 and each c∈(1/2,1), we construct two smooth nonradial symbols of growth at most Ce^c|z|^2 for some constant C>0. Their extended Toeplitz operators in F^2(ℂ) are nonzero and have zero product on all holomorphic polynomials. Moreover, the second symbol is bounded when c≥3/4. Our proofs use Gaussian kernel calculations, matrix identities, theta functions and Fourier transform.
ER  -