Lumer–Phillips theorem

In mathematics, the Lumer–Phillips theorem, named after Günter Lumer and Ralph Phillips, is a result in the theory of strongly continuous semigroups that gives a necessary and sufficient condition for a linear operator in a Banach space to generate a contraction semigroup.

Then A generates a contraction semigroup if and only if[1] An operator satisfying the last two conditions is called maximally dissipative.

Then A generates a contraction semigroup if and only if[2] Note that the condition that D(A) is dense is dropped in comparison to the non-reflexive case.

Then A generates a quasi contraction semigroup if and only if There are many more examples where a direct application of the Lumer–Phillips theorem gives the desired result.

In conjunction with translation, scaling and perturbation theory the Lumer–Phillips theorem is the main tool for showing that certain operators generate strongly continuous semigroups.