Toeplitz operators on the Fock space via the Fourier transform

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.