Products of Toeplitz operators on the Fock space

TL;DR

This paper characterizes when the product of two Toeplitz operators on the Fock space is bounded, revealing that it occurs precisely when the symbols are exponential functions related by a linear polynomial.

Contribution

It provides a complete characterization of bounded products of Toeplitz operators on the Fock space with explicit conditions on their symbols.

Findings

Product of Toeplitz operators is bounded iff symbols are exponential functions of a linear polynomial.

The symbols must be of the form $f(z)=e^{q(z)}$ and $g(z)=ce^{-q(z)}$ with $c eq 0$.

This characterizes the structure of symbols for bounded Toeplitz operator products on the Fock space.

Abstract

Let $f$ and $g$ be functions, not identically zero, in the Fock space $F^2$ of $C_n$. We show that the product $T_fT_{\bar g}$ of Toeplitz operators on $F^2$ is bounded if and only if $f(z)=e^{q(z)}$ and $g(z)=ce^{-q(z)}$, where $c$ is a nonzero constant and $q$ is a linear polynomial.