Dissipative operator

In mathematics, a dissipative operator is a linear operator A defined on a linear subspace D(A) of Banach space X, taking values in X such that for all λ > 0 and all x ∈ D(A)

${\displaystyle \|(\lambda I-A)x\|\geq \lambda \|x\|.}$

A couple of equivalent definitions are given below. A dissipative operator is called maximally dissipative if it is dissipative and for all λ > 0 the operator λI − A is surjective, meaning that the range when applied to the domain D is the whole of the space X.

An operator that obeys a similar condition but with a plus sign instead of a minus sign (that is, the negation of a dissipative operator) is called an accretive operator.

The main importance of dissipative operators is their appearance in the Lumer–Phillips theorem which characterizes maximally dissipative operators as the generators of contraction semigroups.