C0-semigroup

Summary

A $C_0$-semigroup, or strongly continuous semigroup, is a family $T(t)$, $t \ge 0$, of bounded linear operators on a Banach space that behaves like $e^{tA}$. It satisfies $T(0) = I$ and $T(t + s) = T(t) T(s)$, and $T(t) x \to x$ as $t \to 0$ for each vector $x$. Its generator $A$ is in general an unbounded operator, and $u(t) = T(t) u_0$ solves $u' = A u$. $C_0$-semigroups give the solutions of linear evolution equations such as the heat equation, transport equations and Markov processes.