I'm not an analyst, so forgive me if what I'm asking is not suitable for Mathoverflow. For convenience, let $X$ be a compact complex manifold, and $E$ a holomorphic vector bundle on $X$. Let $H$ be the Hilbert space of $L^2$-integrable sections of $E$ on $X$, and let $F$ be a degree zero pseudo-differential operator in $\mathfrak{B}(H)$. Then my question is whether $F$ maps smooth(or complex/real analytic) sections to the same class of functions? What if $F$ is a degree $-1$ pseudo-differential operator?

More specifically, I'm thinking about the case when $E$ is $\oplus _i \wedge ^{0,i} T^*X $ and $F$ is either $\frac{1}{\sqrt{1+D^2}}$ or $\frac{D}{\sqrt{1+D^2}}$, where $D= \bar{\partial}^* + \bar{\partial}$, where $\bar{\partial}$ is the Dolbeault operator.

Any helpful comments and references to learn about this subject is greatly appreciated.

alwaysmaps smooth function to smooth functions. The order tells you how it maps between $L^2$ Sobolev spaces. An operator of order $K$ is a bounded map from $W^{2,j+k}$ to $W^{2,j}$ for any $j$. $\endgroup$ – Deane Yang Mar 12 '18 at 0:36