Zero-product problem for Toeplitz operators on the Fock space

Abstract: We answer Bauer and Le's question on zero products of Toeplitz operators on the Fock space $F^2(\mathbb C<sup>n)$[JFA, 261 (2011), 9, 2617--2640]. For $n\ge2$, we construct two bounded nonradial Schwartz symbols on $\mathbb C<sup>n$ whose Toeplitz operators are nonzero and have zero product on $F^2(\mathbb C<sup>n)$. For $n=1$ and each $c\in(1/2,1)$, we construct two smooth nonradial symbols of growth at most $Ce<sup>{c|z|<sup>2}$ for some constant $C&gt;0$. Their extended Toeplitz operators in $F^2(\mathbb C)$ are nonzero and have zero product on all holomorphic polynomials. Moreover, the second symbol is bounded when $c\geq3/4$. Our proofs use Gaussian kernel calculations, matrix identities, theta functions and Fourier transform.