Dissipative operator

In the linear theory of dissipative operators are linear operators defined on a real or complex Banach spaces which satisfy certain norm estimates. By the theorem of Lumer -Phillips they play an important role in the consideration of strongly continuous semigroups.

Definition

Be a Banach space and. A linear operator with

For all and is called dissipative. This term goes back to Ralph Phillips.

Is a linear operator and dissipative, so called accretive. This term was introduced by Tosio Kato and Kurt Friedrichs.

Hilbert space

If a Hilbert space is a linear operator is dissipative if and only if

Applies in all cases, where the real part called.

Conclusions

Be a dissipative operator on a Banach space.

• Is a surjective iff for all is surjective. Then is called m- dissipative and generates a strongly continuous operator semigroup.
• Is complete if and only if the image of is complete for one.

Example

Looking on a limited area of the Laplace operator with Dirichlet boundary condition (see room ), so we obtain:

The Lax - Milgram proves that is m- dissipative and thus generates a strongly continuous operator semigroup.

• Functional Analysis