Inverse operator of the generator of a $C_0$-semigroup

Let $A$ be the generator of a uniformly bounded $C_0$-semigroup in a Banach space $X$ such that $A$ has a trivial kernel and a dense range. The question whether $A^{-1}$ is a generator of a $C_0$-semigroup is considered. It is shown that the answer is negative in general for $X=\ell^p$, $p\in(1,2)\cup(2,\infty)$. In the case when $X$ is a Hilbert space it is proved that there exist $C_0$-semigroups $(e^{tA})$, $t\geqslant0$, of arbitrarily slow growth at infinity such that the densely defined operator $A^{-1}$ is not the generator of a $C_0$-semigroup.
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