  We prove an extended version of Cordes' lemma concerning trace-class
properties of some special pseudo-differential operators. This version of
Cordes' lemma is used to improve the results in \cite{Arsu} concerning the
Schatten-class properties of pseudo-differential operators in the $(X,\tau)
$-quantization. Here $X$ is an $n$ dimensional vector space and $\tau $ is an
endomorphism of $X$.
