Dissipative operator

On a Hilbert space, an operator $A$ is dissipative when $\mathrm{Re}(Ax,x)\le 0$ for every $x\in D(A)$. This is the infinitesimal form of norm contraction.

A maximal dissipative operator is dissipative and has no proper dissipative extension. Equivalently, for a densely defined dissipative operator, $\lambda I-A$ is onto for some positive $\lambda$.

The Lumer-Phillips theorem says that a densely defined operator generates a contraction C0-semigroup exactly when it is maximal dissipative.