Lumer-Phillips theorem

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

Let be a Banach space and a densely defined , dissipative operator . Then the conclusion of creates a contraction half-group, i.e. for all , exactly when the image of close in lies for one. ${\displaystyle X}$ ${\displaystyle (A,D(A))}$ ${\displaystyle X}$ ${\displaystyle {\overline {A}}}$ ${\displaystyle A}$ ${\displaystyle \|e^{tA}\|\leq 1}$ ${\displaystyle t\geq 0}$ ${\displaystyle \lambda >0}$ ${\displaystyle \lambda -A}$ ${\displaystyle X}$

The theorem was proved in 1961 by Günter Lumer and Ralph Phillips and, together with Hille-Yosida's theorem, is one of the most important theorems in the field of strongly continuous semigroups. In contrast to the Hille-Yosida theorem, however, no estimates are required for the resolvent , so that the application of the Lumer-Phillips theorem in the case of a concrete operator is often easier than the application of the Hille-Yosida theorem.

Inferences

• Let be a tightly defined operator on a Banach space . If both and the adjoint are dissipative, the conclusion of creates a contraction half-group. ${\displaystyle (A,D(A))}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle A^{*}}$ ${\displaystyle A}$
• If a dissipative operator is on a reflexive Banach space and the image is from close in , then the domain of definition of the closure is from close in . From the Lumer-Phillips Theorem it follows that a contraction semigroup creates. ${\displaystyle (A,D(A))}$ ${\displaystyle X}$ ${\displaystyle \lambda -A}$ ${\displaystyle X}$ ${\displaystyle {\overline {A}}}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle {\overline {A}}}$

example

• Considering on (see -space ) the Laplace operator with Dirichlet boundary condition , that is , it is invertible. It also follows from the partial integration . Thus creates a contraction half group. ${\displaystyle X:=L^{2}([0,1],\mathbb {R} )}$ ${\displaystyle L^{p}}$ ${\displaystyle Au:=u''}$ ${\displaystyle D(A):=\{u\in H^{2}([0,1],\mathbb {R} ):u(0)=u(1)=0\}}$ ${\displaystyle A:D(A)\rightarrow X}$ ${\displaystyle \langle Au,u\rangle =\int _{0}^{1}u''u\,\mathrm {d} x=-\int _{0}^{1}u'u'\,\mathrm {d} x\leq 0}$ ${\displaystyle A}$

literature

• Ammon Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations . Applied Mathematical Sciences 44 , Springer-Verlag, Berlin 1983, ISBN 3-540-90845-5 .
• Klaus-Jochen Engel, Rainer Nagel: One-Parameter Semigroups for linear Evolution Equations . Graduate Texts in Mathematics 194, Springer-Verlag 2000, ISBN 0-387-98463-1 .