Let $(X,\|\cdot\|_X)$ be a [Banach space](/page/Banach%20Space). Let $I_X:X\to X$ denote the identity map, and let $\mathcal{L}(X)$ denote the Banach space of bounded linear maps $X\to X$ with operator norm $\|\cdot\|_{\mathcal{L}(X)}$. Let $(T(t))_{t\geq 0}$ be a strongly continuous semigroup on $X$, meaning $T(t)\in\mathcal{L}(X)$ for every $t\geq 0$, $T(0)=I_X$, $T(t+s)=T(t)T(s)$ for all $s,t\geq 0$, and $t\mapsto T(t)x$ is continuous as a map $[0,\infty)\to X$ for every $x\in X$. Let $A:D(A)\subset X\to X$ be the infinitesimal generator of $(T(t))_{t\geq 0}$, where $D(A)\subset X$ is defined by

\begin{align*} D(A)=\left\{x\in X: \lim_{h\downarrow 0}\frac{T(h)x-x}{h}\text{ exists in }X\right\}. \end{align*}

For every $x\in D(A)$, define $Ax\in X$ by

\begin{align*} Ax=\lim_{h\downarrow 0}\frac{T(h)x-x}{h}. \end{align*}

Let $u_0\in D(A)$, and define $u:[0,\infty)\to X$ by

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

Then, for every $S>0$, the restriction $u|_{[0,S]}$ is a classical solution of the homogeneous [abstract Cauchy problem](/page/Abstract%20Cauchy%20Problem) with initial condition $u(0)=u_0$, meaning $u(t)\in D(A)$ for every $t\in[0,S]$, $u:[0,S]\to X$ is continuous, $u:(0,S)\to X$ is differentiable, $u'(t)=Au(t)$ for every $t\in(0,S)$, $t\mapsto Au(t)$ is continuous as a map $[0,S]\to X$, and the right derivative at $0$ exists and equals $Au_0$.