Let $n \in \mathbb{N}$, let $m \in \mathbb{R}$, let $\mathbb{N}_0 := \mathbb{N} \cup \{0\}$, and let $a \in S^m_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$ be a complex-valued Kohn-Nirenberg symbol. Let $\mathcal{S}'(\mathbb{R}^n)$ denote the tempered-distribution dual of the [Schwartz space](/page/Schwartz%20Space) $\mathcal{S}(\mathbb{R}^n)$, and let

\begin{align*} A = \operatorname{Op}(a): \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^n) \end{align*}

be the global Kohn-Nirenberg operator defined by

\begin{align*} (\operatorname{Op}(a)u)(x) = (2\pi)^{-n}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} e^{i(x-y)\cdot \xi} a(x,\xi)u(y)\,d\mathcal{L}^n(y)\,d\mathcal{L}^n(\xi). \end{align*}

Let $A^*: \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)$ denote the formal adjoint of $A$ with respect to the standard $L^2(\mathbb{R}^n)$ pairing. If $\sigma_m(A)$ denotes the class of $a$ in the quotient $S^m_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)/S^{m-1}_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$, then $A^* \in \Psi^m_{1,0}(\mathbb{R}^n)$ and

\begin{align*} \sigma_m(A^*) = \overline{\sigma_m(A)}. \end{align*}