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 $F^2({\mathbb{C}}^n)$[JFA, 261 (2011), 9, 2617--2640]. For $n\ge 2$, we construct two bounded nonradial Schwartz symbols on ${\mathbb{C}}^n$ whose Toeplitz operators are nonzero and have zero product on $F^2({\mathbb{C}}^n)$. For $n=1$ and each $c\in (1/2,1)$, we construct two smooth nonradial symbols of growth at most $C{e}^{c|z{|}^2}$ for some constant $C>0$. 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 $c\ge 3/4$. Our proofs use Gaussian kernel calculations, matrix identities, theta functions and Fourier transform.

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