Zero-product problem for arbitrary bounded Bergman symbols

Determine whether two arbitrary bounded measurable symbols on the Bergman space of the unit disk must satisfy f=0 or g=0 almost everywhere whenever their Toeplitz operators satisfy T_fT_g=0.

Background

The paper reviews known zero-product results for restricted classes of Bergman-space symbols, including bounded harmonic symbols, symbols with one radial factor, and bounded pluriharmonic symbols. It then identifies the unresolved case of two arbitrary bounded symbols, even in the Bergman space of the unit disk.

References

For two arbitrary bounded symbols, the zero product problem remains open even on the unit disk.

— Zero-product problem for Toeplitz operators on the Fock space  (2609.20555 - Qin, 17 Sep 2026) in Section 1, paragraph discussing Bergman-space results