Suppose I have a self-adjoint pseudo-differential operator $A$ on $\mathbb{R}^n$ and a continuous function $f$ (possibly bounded, or Schwartz, or compactly supported) on its spectrum. Then I can consider the operator $f(A)$ defined by functional calculous.


Is $f(A)$ again a pseudo-differential operator and if yes, how are the symbols related?


In what way does the type of operator or the type of function matter?



Functions of pseudodifferential operators
Here is a good reference for this



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> Michael Taylor: Pseudodifferential Operators, Princeton University PRess, 1981
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In Chapter 12 it explains how to construct $f(A)$ when $A$ is elliptic selfadjoint of order $1$, $A\geq 0$, and $f$ is a smooth symbol of order $m$, i.e., $f$ is smooth and 


$$ f^{(k)}(\lambda)= O(|\lambda|^{m-k}),\;\;\lambda\to\infty $$,


for any nonnegative integer $k$ 


