Dissipative operator explained

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)

\|( λ I - A)x\|\geq λ \|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.^[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

\|( λ I - A)x\|\ne 0 , 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 \|( λ I - A)x\|> λ \|x\| for all non-null x in the domain, then, by the triangle inequality, \| λ x\| + \|Ax\|\ge\|( λ I - A)x\|> λ \|x\|, which implies that A itself has an inverse.) We may then state that\|( λ I - A) ^{-1} z\|\leq (1)/(λ) \|z\|

for all z in the range of λI − A. This is the same inequality as that given at the beginning of this article, with

z = ( λ I - A)x. (We could equally well write these as \|(I - \kappa A) ^{-1} z\|\leq\|z\| or \|(I - \kappa A)x\|\geq\|x\| 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

J(x): = \left\{x'\in X':\|x'\| ^{2= \langle} _{X} x',x\rangle \right\}.

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

{\rm Re}\langle Ax,x'\rangle\leq 0 .

In the case of Hilbert spaces, this becomes

{\rm Re}\langle Ax,x\rangle\leq 0 for all x in D(A). Since this is non-positive, we have\|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} \therefore\|x - Ax\|\geq\|x + Ax\|

Since I−A has an inverse, this implies that

(I + A)(I - A) ^{-1} is a contraction, and more generally, ( λ I + A)( λ I - A) ^{-1} 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.

Examples

• For a simple finite-dimensional example, consider n-dimensional Euclidean space Rⁿ with its usual dot product. If A denotes the negative of the identity operator, defined on all of Rⁿ, then

x ⋅ A x = x ⋅ ( - x) = - \| x \| ^{2} \leq 0 ,

so A is a dissipative operator.

• So long as the domain of an operator A (a matrix) is the whole Euclidean space, then it is dissipative if and only if A+A^* (the sum of A and its adjoint) does not have any positive eigenvalue, and (consequently) all such operators are maximally dissipative. This criterion follows from the fact that the real part of

x ^{*} Ax, which must be nonpositive for any x, is x ^{*} (A + A ^{*})/(2 x.) The eigenvalues of this quadratic form must therefore be nonpositive. (The fact that the real part of x ^{*} Ax, must be nonpositive implies that the real parts of the eigenvalues of A must be nonpositive, but this is not sufficient. For example, if A = \begin{pmatrix} -1 & 3 \\ 0 & -1 \end{pmatrix} then its eigenvalues are negative, but the eigenvalues of A+A^* are −5 and 1, so A is not dissipative.) An equivalent condition is that for some (and hence any) positive λ , λ- A has an inverse and the operator ( λ+ A)( λ- A) ^{-1} is a contraction (that is, it either diminishes or leaves unchanged the norm of its operand). If the time derivative of a point x in the space is given by Ax, then the time evolution is governed by a contraction semigroup that constantly decreases the norm (or at least doesn't allow it to increase). (Note however that if the domain of A is a proper subspace, then A cannot be maximally dissipative because the range will not have a high enough dimensionality.)H ^{1([0, 1]; R)} with u(1) = 0. D(A) is dense in L²([0, 1]; R). Moreover, for every u in D(A), using integration by parts,\langle u, A u \rangle = \int ^{1} _{0} u(x) u'(x) d x = - (1 { 2 })/(u( 0 )) ^{2} \leq 0 .

Hence, A is a dissipative operator. Furthermore, since there is a solution (almost everywhere) in D to

u -λ u' = f for any f in H, the operator A is maximally dissipative. Note that in a case of infinite dimensionality like this, the range can be the whole Banach space even though the domain is only a proper subspace thereof.

• Consider H = H₀²(Ω; R) (see Sobolev space) for an open and connected domain Ω ⊆ Rⁿ and let A = Δ, the Laplace operator, defined on the dense subspace of compactly supported smooth functions on Ω. Then, using integration by parts,

\langle u, \Delta u \rangle = \int _{\Omega} u(x) \Delta u(x) d x = - \int _{\Omega} | \nabla u(x) | ^{2} d x = - \| \nabla u \| ^{2} _{L ^{2} (\Omega; R )} \leq 0 ,

so the Laplacian is a dissipative operator.

References

• Book: Engel. Klaus-Jochen. Nagel. Rainer . One-parameter semigroups for linear evolution equations . 2000. Springer.
• Book: Renardy, Michael . Rogers, Robert C.. An introduction to partial differential equations. Texts in Applied Mathematics 13. Second. Springer-Verlag. New York. 2004. 356. 0-387-00444-0. (Definition 12.25)

Notes and References

1. Encyclopedia: Dissipative operator. Encyclopedia of Mathematics.
2. Engel and Nagel Proposition II.3.14
3. The theorem implies that for a given x there exists a continuous linear functional φ with the property that φ(x)=‖x‖, with the norm of φ equal to 1. We identify ‖x‖φ with x'.
4. Engel and Nagel Exercise II.3.25i
5. Engel and Nagel Example II.3.26
6. Engel and Nagel Proposition II.3.23