Dissipative operator

Summary

A dissipative operator is a linear operator $A$ on a Banach space such that $\norm{(\lambda - A)x} \ge \lambda \norm{x}$ for every $\lambda > 0$. On a Hilbert space this means $\operatorname{Re}\ip{Ax}{x} \le 0$ for every $x$ in the domain. The densely defined maximal dissipative operators are exactly the generators of semigroups that do not increase norms, by the Lumer–Phillips theorem. The name comes from physics: the evolution $u' = Au$ loses energy and never gains it.