Contraction semigroup

Summary

A contraction semigroup is a strongly continuous semigroup $(T(t))_{t \ge 0}$ of operators on a Banach space with $\norm{T(t)} \le 1$ for every $t \ge 0$. It describes a linear evolution that never increases the norm of the state. The Lumer–Phillips theorem says that its generators are exactly the densely defined m-dissipative operators. Heat flows, diffusions and unitary groups restricted to $t \ge 0$ give contraction semigroups.