Let $X$ be a [Banach space](/page/Banach%20Space) and let $A: D(A) \subset X \to X$ be a densely defined linear operator. The homogeneous [abstract Cauchy problem](/page/Abstract%20Cauchy%20Problem) for $A$ is well-posed in the semigroup sense on $X$ if and only if $A$ generates a strongly continuous semigroup $(T(t))_{t \ge 0}$ with $T(t): X \to X$ for every $t \ge 0$ and the mild solution with initial value $u_0 \in X$ is given by

\begin{align*} u(t)=T(t)u_0, \qquad t \ge 0. \end{align*}