Let $X$ be a [Banach space](/page/Banach%20Space). Suppose that the homogeneous autonomous [abstract Cauchy problem](/page/Abstract%20Cauchy%20Problem) on $X$ has the following properties.

For every $u_0 \in X$, there exists a unique admissible solution $u(\cdot;u_0): [0,\infty) \to X$ in a solution class stable under time translation, with initial condition $u(0;u_0)=u_0$. More precisely, if $s \ge 0$ and $u(\cdot;u_0)$ is the admissible solution with initial value $u_0$, then the shifted map $v_s: [0,\infty) \to X$ defined by $v_s(r)=u(r+s;u_0)$ is an admissible solution of the same autonomous problem with initial value $v_s(0)=u(s;u_0)$.

Assume further that, for every $t \ge 0$, the map $u_0 \mapsto u(t;u_0)$ is bounded and linear from $X$ to $X$, and that, for every $u_0 \in X$, the trajectory $t \mapsto u(t;u_0)$ is continuous from $[0,\infty)$ to $X$.

For each $t \ge 0$, define $T(t): X \to X$ by

\begin{align*} T(t)u_0 = u(t;u_0). \end{align*}

Then $T(0)=I_X$, $T(t+s)=T(t)T(s)$ for all $s,t \ge 0$, each $T(t)$ belongs to $\mathcal{L}(X)$, and $(T(t))_{t \ge 0}$ is a strongly continuous semigroup on $X$.