Dissipative operator

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 xD(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 λIA 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]

for all z in the range of λIA. 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.)

Equivalent characterizations

Define the duality set of xX, a subset of the dual space X' of X, by

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 xD(A) there exists a x'J(x) such that

In the case of Hilbert spaces, this becomes for all x in D(A). 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 (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

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

Notes

  1. "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

References

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