Integral formulas for the Weyl and anti-Wick symbols

Abstract

The first purpose of this article is to provide conditions for a bounded operator in $L^2({\mathbb{R}}^n)$ to be the Weyl (resp. anti-Wick) quantization of a bounded continuous symbol on ${\mathbb{R}}^{2n}$. Then, explicit formulas for the Weyl (resp. anti-Wick) symbol are proved. Secondly, other formulas for the Weyl and anti-Wick symbols involving a kind of Campbell Hausdorff formula are obtained. A point here is that these conditions and explicit formulas depend on the dimension $n$ only through a Gaussian measure on ${\mathbb{R}}^{2n}$ of variance $1/2$ in the Weyl case (resp. variance $1$ in the anti-Wick case) suggesting that the infinite dimension setting for these issues could be considered. Besides, these conditions are related to iterated commutators recovering in particular the Beals characterization Theorem.