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. 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. Properties A dissipative operator has the following properties: From the inequality given above, we see that for any x in the domain of A, if ‖x‖ ≠ 0 then so the kernel of λI − A is just the zero vector and λI − A is therefore injective and has an inverse for all λ > 0. (If we have the strict inequality for all non-null x in the domain, then, by the triangle inequality, which implies that A itself has an inverse.) We may then state that for all z in the range of λI − A. This is the same inequality as that given at the beginning of this article, with (We could equally well write these as which must hold for any positive κ.) λI − A is surjective for some λ > 0 if and only if it is surjective for all λ > 0. (This is the aforementioned maximally dissipative case.) In that case one has (0, ∞) ⊂ ρ(A) (the resolvent set of A). A is a closed operator if and only if the range of λI - A is closed for some (equivalently: for all) λ > 0. Equivalent characterizations Define the duality set of x ∈ X, a subset of the dual space X of X, by By the Hahn–Banach theorem this set is nonempty. In the Hilbert space case (using the canonical duality between a Hilbert space and its dual) it consists of the single element x. More generally, if X is a Banach space with a strictly convex dual, then J(x)