Some conditions for a $C_0$-semigroup to be asymptotically finite-dimensional

We study the class of bounded $C_0$-semigroups $\mathscr{T}=(T_t)_{t\geqslant0}$ on a Banach space $X$ satisfying the asymptotic finite dimensionality condition: $\operatorname{codim}X_0(\mathscr{T})<\infty$, where $X_0(\mathscr{T}):=\{x\in X:\lim\limits_{t\to\infty}\|T_tx\|=0\}$. We prove a theorem which provides some necessary and sufficient conditions for asymptotic finite dimensionality.