This is the abstract of the paper
  "Some Remarks about the Perron Condition for $C_0$-Semigroups"
by Constantin Buse.

For the whole paper send the command

  get evolve-l 98-0014

to: "listserv@rz.uni-karlsruhe.de".

Instead you may get the paper at
  "http://ma1serv.mathematik.uni-karlsruhe.de/evolve-l/index.html"
too.

Abstract:
We prove the following. Let ${\bf T}=\{T(t)\}_{t\ge 0}$ be a bounded $C_0$-
semigroup on a Banach space $X$ and $A$ its infinitesimal generator. Then
$\Re \sigma(A)<0$ if and only if
$$\sup\limits_{t>0}||\int\limits_0^te^{i\mu\xi}T(\xi)d\xi||<\infty,\quad\forall
\mu\in{\bf R},\forall x\in X.$$
In particular we obtain that a strongly continuous and exponentially bounded
family of bounded linear operators ${\cal U}$ on $X$ is uniformly exponentially
stable if and only if the spectrum of the infinitesimal generator of the
evolution semigroup associated to ${\cal U},$ is lying in ${\bf C}_-:=\{
z\in{\bf C} : \Re(z)<0\}.$