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?