Let $$\Omega$$ be an open and connect subset of $$\mathbb{R}^2$$,we denote by $$\partial \Omega$$ its boundary the latter is supposed to be smooth ($$\mathcal{C}^\infty)$$, its outword normal vector is denoted by $$n$$. Let $$f: \Omega \mapsto \mathbb{R}$$ such that $$f(x)\geq \alpha > 0$$. Now, let $$A : \mathbb{H}^{1/2}(\partial \Omega)\mapsto \mathbb{H}^{-1/2}(\partial \Omega)$$. Such that $$A(\varphi)=\frac{\partial u }{n}$$, with $$u$$ is the unique solution in $$\mathbb{H}^1(\Omega)$$ of
$$div(f\nabla u)=0$$ and $$u_{|\partial\Omega}=\varphi$$.
Can I say that $$A$$ is a pseudo-differential Operator ?
Finally, The answer is yes. In deed the result remains true if we remplace $$div(f \nabla .)$$by any other second order elliptic opertor. Morover, $$A$$ is of order one. The proof can be found here : https://arxiv.org/pdf/1212.6785.pdf