Schatten-von Neumann properties in the Weyl calculus.

\begin{abstract} Let $\mathrm{Op} _t(a)$, for $t\in \mathbf R$, be the pseudo-differential operator $$ f(x) \mapsto (2\pi )^{-n}\iint a((1-t)x+ty,\xi )f(y)e^{i\langle x-y,\xi\rangle\, dyd\xi $$ and let $\mathscr I_p$ be the set of Schatten-von Neumann operators of order $p\in [1,\infty ]$ on $L^2$. We are especially concerned with the Weyl case (i.{\,}e. when $t=1/2$). We prove that if $m$ and $g$ are appropriate metrics and weight functions respectively, $h_g$ is the Planck's function, $h_g^{k/2}m\in L^p$ for some $k\ge 0$ and $a\in S(m,g)$, then $\\mathrm{Op} _t(a)\in \mathscr I_p$, iff $a\in L^p$. Consequently, if $0\le \delta <\rho \le 1$ and $a\in S^r_{\rho ,\delta}$, then $\mathrm{Op} _t(a)$ is bounded on $L^2$, iff $a\in L^\infty$. \end{abstract}