In this talk, we present new results on the characterization of strongly stable semigroups in terms of the spectral properties of their infinitesimal generators. Until now, spectral characterizations applicable to arbitrary semigroups on general Banach spaces have remained unknown. Within the framework of the Abstract Cauchy Problem (ACP),
\[
\left\{
\begin{aligned}
\dot{u}(t) &= Au(t), \quad t\geq0, \\
u(0) &= x.
\end{aligned}
\right.
\]
for a given $x\in X$, such characterizations are particularly important for deriving stability properties of individual solutions without explicitly having to determine them.

To tackle this problem, we introduce a new spectral concept for operators, which we called the pseudofunction spectrum, and employ Tauberian methods. Using this notion, we obtain general characterization results and provide simple quick proofs of several classical sufficient criteria for strong stability, such as the Arendt-Batty-Lyubuch-Vu theorem.

This work is based on joint research with L. Neyt and J. Vindas.