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?
Here is a good reference for this
Michael Taylor: Pseudodifferential Operators, Princeton University PRess, 1981
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 $k$
Another reference is
Dimassi-Sjöstrand: Spectral asymptotics in the semiclassical limit, Cambridge University Press (1999)
See in particular Chapter 8 ("Functional Calculus for pseudodifferential operators").
The results are in the semiclassical setting. The result which may interest you is Theorem 8.7 which states that if $P$ is a selfsadjoint h-pseudodiff operator and $f\in C^\infty_0$, then f(P) is again a pseudo with leading symbol $f(p((x,\xi))$ , where $p(x,\xi)$ is the leading symbol of $P$.
The method to prove this is based here on a standard Cauchy formula. (I think in the book of Taylor, cited by Liviu Nicolaescu a different strategy is used).
Since the question posed is about the "In what way does the type of operator or the type of function matter?", I thought the following observation will be apt:
As pointed out by Liviu Nicolaescu in the comment above, Taylor's approach seems to have much wider applicability when it comes to functional calculus. In fact in page 295 of Taylor's book it is mentioned that Seeley's results form a special case of the result.
Moreover, these methods have gone beyond elliptic operators. For instance, Uhlmann, Melrose and Guillemin have developed a framework of distributions whose wavefront sets are in several Lagrangian intersecting manifolds (pseudodifferential operators with singular symbols) for a functional calculus on real principal-type operators, operators of double characteristics, wave operators. Principal symbols also have been computed for these operators and all the computations are purely symbolic.
