Lumer–Phillips theorem

The set of Lumer -Phillips is a result from the theory of strongly continuous semigroups and characterized contraction semigroups:

Be a Banach space and in densely defined dissipative operator. Then, the conclusion of a contraction semigroup, ie for all if and only if is a picture of dense in.

The theorem was proved in 1961 by Günter Lumer and Ralph Phillips and belongs to the set of Hille - Yosida to the most important records in the field of strongly continuous semigroups. In contrast to the set of Hille - Yosida but no estimates for the resolvent are needed so that the application of the theorem of Lumer -Phillips in the case of a concrete operator is often simpler than the application of the theorem of Hille - Yosida.

Conclusions

• Be a densely defined operator on a Banach space. Are both the adjoint dissipative, produces the conclusion of a contraction semigroup.
• If a dissipative operator on a reflexive Banach space and is the image of dense in, then the domain of definition to the conclusion of is dense in. It follows from the theorem of Lumer -Phillips that a contraction semigroup generated.

Example

• If one considers (see room ) the Laplace operator with Dirichlet boundary condition, ie, it is invertible. Moreover, it follows from partial integration. Thus generates a contraction semigroup.