Let G be a step two nilpotent Lie group. In this paper, we give necessary and sufficient conditions on the operator valued symbols σ such that the associated pseudo-differential operators $T_{\sigma }$ on G are in the class of Hilbert-Schmidt operators. As a key step to prove this, we define $(\mu ,\nu )$-Weyl transform on G and derive a trace formula for $(\mu ,\nu )$-Weyl transform with symbols in $L^2\left({\mathbb{R}}^{2n}\right)$. We show that Hilbert-Schmidt pseudo-differential operators on $L^2(G)$ are same as Hilbert-Schmidt $(\mu ,\nu )$-Weyl transform with symbol in $L^2({\mathbb{R}}^{2n+r+k}\times {\mathbb{R}}^{2n+r+k})$. Further, we present a characterization of the trace class pseudo-differential operators on G and provide a trace formula for these trace class operators.

设G是一个二阶幂零李群。在本文中，我们给出了算子值符号σ 的充要条件，使得相关的伪微分算子${吨}_{\sigma }$上ģ是在类希尔伯特-施密特运营商。作为证明这一点的关键步骤，我们定义$(\mu ,\nu )$-Weyl 对G 进行变换并推导出迹公式$(\mu ,\nu )$-Weyl 变换与符号 ${升}^2\left({\mathbb{电阻}}^{2n}\right)$. 我们证明了 Hilbert-Schmidt 伪微分算子${升}^2(G)$ 与希尔伯特-施密特相同 $(\mu ,\nu )$-Weyl 变换，带符号 ${升}^2({\mathbb{电阻}}^{2n+r+克}\times {\mathbb{电阻}}^{2n+r+克})$. 此外，我们提出了G上迹类伪微分算子的表征，并为这些迹类算子提供了迹公式。