I am considering the Cauchy IVP for the evolution equation $$u_t + \Psi u =0$$ where $\Psi$ is a linear pseudo differential operator with symbol $\widehat{\Psi}\left(\underline{\xi}\right)$.

The corresponding solution semigroup is $$u = \mathcal{F}^{-1}\left( e^{-\widehat{\Psi} t} \mathcal{F}(f) \right)$$

What are the conditions under which a linear pseudo-differential operator generates a strongly continuous semigroup in terms of its symbol? I was thinking there would be a way in which the Hille-Yosida conditions could be rewritten in terms of symbols rather than operators.

Thanks :)