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\|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d70faafc3e9e9c1992a87afe9a9fee7cf829eda)
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.[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.
Properties
A dissipative operator has the following properties:[2]
- 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
![{\displaystyle \|(\lambda I-A)^{-1}z\|\leq {\frac {1}{\lambda ))\|z\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd548e3d662b0363ce764478b11dc27fdf1ed6bc)
- 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
![{\displaystyle J(x):=\left\{x'\in X':\|x'\|_{X'}^{2}=\|x\|_{X}^{2}=\langle x',x\rangle \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/277c915eb41b0dfbd57b87441ba357aaec7f5d4d)
By the Hahn–Banach theorem this set is nonempty.[3] In the Hilbert space case (using the canonical duality between a Hilbert space and its dual) it consists of the single element x.[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
![{\displaystyle {\rm {Re))\langle Ax,x'\rangle \leq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce7bad9cd0efc00da3468f3232c1b0cd98c238e4)
In the case of Hilbert spaces, this becomes
for all x in D(A). Since this is non-positive, we have
![{\displaystyle \|x-Ax\|^{2}=\|x\|^{2}+\|Ax\|^{2}-2{\rm {Re))\langle Ax,x\rangle \geq \|x\|^{2}+\|Ax\|^{2}+2{\rm {Re))\langle Ax,x\rangle =\|x+Ax\|^{2))](https://wikimedia.org/api/rest_v1/media/math/render/svg/006be6a7e6209c88476847f24cc06571490e6e1b)
![{\displaystyle \therefore \|x-Ax\|\geq \|x+Ax\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a373231b302462a88f591e2934dfff2a5d5dfcc1)
Since I−A has an inverse, this implies that
is a contraction, and more generally,
is a contraction for any positive λ. 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.