Math 6410 • ODE

Let \(\{T(t):t\ge 0\}\) be a strongly continuous semigroup on \(H\) with infinitesimal generator \(A\). Show that \[ T(t):D(A)\longrightarrow D(A) \] and that the map \[ t\longmapsto T(t)x,\qquad t\in [0,\infty ), \] is differentiable and \[ \frac {d}{dt}T(t)x=AT(t)x=T(t)Ax,\quad x\in D(A). \]