Let $m \in \mathbb{R}$, let $n \in \mathbb{N}$, and let $a: \mathbb{R}^n_x \times \mathbb{R}^n_\xi \to \mathbb{C}$ be a Hörmander symbol in $S^m_{1,0}(\mathbb{R}^n_x \times \mathbb{R}^n_\xi)$. Let $A := \operatorname{Op}(a): \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)$ be the Kohn-Nirenberg quantisation defined, for $u \in \mathcal{S}(\mathbb{R}^n)$, by the oscillatory integral

\begin{align*} Au(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*}

The formal adjoint $A^*: \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)$ is defined by

\begin{align*} \int_{\mathbb{R}^n} Au(x)\overline{v(x)}\, d\mathcal{L}^n(x) = \int_{\mathbb{R}^n} u(y)\overline{A^*v(y)}\, d\mathcal{L}^n(y) \end{align*}

for all $u,v \in \mathcal{S}(\mathbb{R}^n)$. Use the pseudodifferential convention

\begin{align*} D_x^\alpha := \left(\frac{1}{i}\right)^{|\alpha|}\partial_x^\alpha, \qquad D_\xi^\alpha := \left(\frac{1}{i}\right)^{|\alpha|}\partial_\xi^\alpha \end{align*}

for each multi-index $\alpha \in \mathbb{N}_0^n$. Then there exists a symbol $a^*: \mathbb{R}^n_x \times \mathbb{R}^n_\xi \to \mathbb{C}$ in $S^m_{1,0}(\mathbb{R}^n_x \times \mathbb{R}^n_\xi)$ such that $A^* - \operatorname{Op}(a^*)$ is smoothing, meaning it has a $C^\infty$ Schwartz kernel and maps $\mathcal{S}'(\mathbb{R}^n)$ continuously into $C^\infty(\mathbb{R}^n)$. The symbol $a^*$ has the asymptotic expansion

\begin{align*} a^*(x,\xi) \sim \sum_{\alpha \in \mathbb{N}_0^n}\frac{1}{\alpha!}\partial_\xi^\alpha D_x^\alpha \overline{a(x,\xi)} \end{align*}

in the sense that, for every $N \in \mathbb{N}$,

\begin{align*} a^*(x,\xi) - \sum_{|\alpha|<N}\frac{1}{\alpha!}\partial_\xi^\alpha D_x^\alpha \overline{a(x,\xi)} \in S^{m-N}_{1,0}(\mathbb{R}^n_x \times \mathbb{R}^n_\xi). \end{align*}

In particular,

\begin{align*} a^* - \overline{a} \in S^{m-1}_{1,0}(\mathbb{R}^n_x \times \mathbb{R}^n_\xi). \end{align*}