Generator of a semigroup

Summary

The generator of a strongly continuous semigroup $(T(t))_{t \ge 0}$ of operators is the derivative of $T(t)$ at $t = 0$. It is usually an unbounded, closed, densely defined operator $A$. The semigroup solves the abstract Cauchy problem $u'(t) = Au(t)$, $u(0) = x$, through $u(t) = T(t)x$, and one writes $T(t) = e^{tA}$. The generator determines the semigroup uniquely, so questions about evolution equations become questions about one operator.