Toeplitz operators on the Fock space via the Fourier transform

TL;DR

This paper uses the Fourier transform to decompose Toeplitz operators on the Fock space, providing new conditions for boundedness and compactness, and applying the theory to Schatten norm estimates of operator products.

Contribution

It introduces a Fourier transform-based decomposition of Toeplitz operators, linking symbol properties to operator boundedness, compactness, and Schatten norm estimates.

Findings

Decomposition of Toeplitz operators into sums with compactly supported symbols.

Sufficient conditions for boundedness based on Carleson measure criteria.

Estimation of Schatten p-norms for products of Toeplitz operators.

Abstract

In sprite by Berger-Coburn theorems and their conjecture in \cite{Coburn1994}, we use the Fourier transform to decompose $ T_{g}$ as an infinite sum of Toeplitz operators with symbols which have compact support in the frequency domain. As a consequence, we obtain a sufficient condition for $ T_{g}$ to be bounded in terms of the Carleson measure conditions defined by the heat transform of the symbol $g$. Moreover the decomposition of a Toeplitz operator leads us to get easily understanding that for a bounded function $g$, if its Berezin transform vanishes at infinity, then the Toeplitz operator $T_g$ is compact \cite{Eng} and the Toeplitz algebra generated by Toeplitz operators with symbols in $L^{\infty}$ is indeed generated by Toeplitz operators with symbols which on uniformly continuous on ${\mathbb C}^n$ \cite{Bauer2012}.Further, we will apply our decomposition theory for a Toeplitz operator to estimate the Schatten $p$-norm of the product of two Toeplitz operators.