3 added 9 characters in body

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 $k$

2 clarified exposition

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 selfadjoint of order $1$, $A\geq 0$, and $f$ is a smooth symbol of order $m$.m$, i.e.,$f$is smooth and $$f^{(k)}(\lambda)= O(|\lambda|^{m-k}),\;\;\lambda\to\infty$$, for any nonnegative integer$k$1 Here is a good reference for this Michael Taylor: Pseudodifferential Operators, Princeton University PRess, 1981 In Chapter 12 it explains how to construct$f(A)$when$A$is selfadjoint of order$1$,$A\geq 0$, and$f$is a symbol of order$m\$.