Let $n \in \mathbb{N}$ with $n \geq 1$, let $r \in \mathbb{R}$, let $\mathcal{L}^n$ denote $n$-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}^n$, and let $\mathbb{R}^n_0 := \mathbb{R}^n \setminus \{0\}$. Let $B:C_c^\infty(\mathbb{R}^n) \to C_c^\infty(\mathbb{R}^n)$ be a properly supported pseudodifferential operator in $\Psi^r_{1,0}(\mathbb{R}^n)$. Let $B^*:C_c^\infty(\mathbb{R}^n) \to C_c^\infty(\mathbb{R}^n)$ denote the formal adjoint of $B$ with respect to the Lebesgue $L^2$ [inner product](/page/Inner%20Product) on $\mathbb{R}^n$ defined using $\mathcal{L}^n$. Assume the standard properly supported $\Psi_{1,0}$ pseudodifferential calculus: every properly supported operator in $\Psi^a_{1,0}(\mathbb{R}^n)$ maps $C_c^\infty(\mathbb{R}^n)$ into $C_c^\infty(\mathbb{R}^n)$, the formal adjoint $B^*$ is properly supported and belongs to $\Psi^r_{1,0}(\mathbb{R}^n)$, the composition of properly supported operators in $\Psi^a_{1,0}(\mathbb{R}^n)$ and $\Psi^b_{1,0}(\mathbb{R}^n)$ belongs to $\Psi^{a+b}_{1,0}(\mathbb{R}^n)$, the principal symbol class of a formal adjoint is represented by the pointwise complex conjugate of any representative of the original principal symbol class, and the principal symbol class of a composition is represented by the pointwise product of representatives of the principal symbol classes of the factors. Then $B^*B \in \Psi^{2r}_{1,0}(\mathbb{R}^n)$. Moreover, for every $u \in C_c^\infty(\mathbb{R}^n)$, $(B^*Bu,u)_{L^2(\mathbb{R}^n)} = \|Bu\|_{L^2(\mathbb{R}^n)}^2 \geq 0$. If $b_r:\mathbb{R}^n \times \mathbb{R}^n_0 \to \mathbb{C}$ is any representative of the principal symbol class $\sigma_r(B)$, then the principal symbol class $\sigma_{2r}(B^*B)$ is represented by the map $c_{2r}:\mathbb{R}^n \times \mathbb{R}^n_0 \to \mathbb{C}$ given by $c_{2r}(x,\xi)=|b_r(x,\xi)|^2$ for every $(x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n_0$.