2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/117940We 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,τ) $-quantization. Here $X$ is an $n$ dimensional vector space and $τ$ is an endomorphism of $X$.Functional AnalysisAnalysis of PDEs35S05, 43Axx, 46-XX, 47-XX, 42B15, 42B35text