Dissipative operator

In mathematics, a dissipative operator is a linear operator A defined on a linear subspace D of Banach space X, taking values in X such that for all λ > 0 and all x ∈ D
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 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. We may then state thatλI − A is surjective for some λ > 0 if and only if it is surjective for all λ > 0. In that case one has ⊂ ρ.A is a closed operator if and only if the range of λI - A is closed for some 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 it consists of the single element x. More generally, if X is a Banach space with a strictly convex dual, then J consists of a single element.
Using this notation, A is dissipative if and only if for all x ∈ D there exists a x' ∈ J such that
In the case of Hilbert spaces, this becomes for all x in D. Since this is non-positive, we have
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, in contrast to ⁻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
So long as the domain of an operator A is the whole Euclidean space, then it is dissipative if and only if A+''A^* does not have any positive eigenvalue, and all such operators are maximally dissipative. This criterion follows from the fact that the real part of which must be nonpositive for any x'', is The eigenvalues of this quadratic form must therefore be nonpositive. An equivalent condition is that for some positive has an inverse and the operator is a contraction. 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.
Consider H = L² with its usual inner product, and let Au = u′ with domain D equal to those functions u in the Sobolev space with u = 0. D is dense in L². Moreover, for every u in D, using integration by parts,
Consider H = H₀² 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,