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) A couple of equivalent definitions are given below.
[1] 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.
[4] More generally, if X is a Banach space with a strictly convex dual, then J(x) consists of a single element.
[5] Using this notation, A is dissipative if and only if[6] for all x ∈ D(A) there exists a x' ∈ J(x) such that In the case of Hilbert spaces, this becomes
The utility of this formulation is that if this operator is a contraction for some positive λ then A is dissipative.
It is not necessary to show that it is a contraction for all positive λ (though this is true), in contrast to (λI−A)−1 which must be proved to be a contraction for all positive values of λ.