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.

Statement of the theorem

Let A be a linear operator defined on a linear subspace D(A) of the Banach space X. Then A generates a contraction semigroup if and only if An operator satisfying the last two conditions is called maximally dissipative.

Variants of the theorem

Reflexive spaces

Let A be a linear operator defined on a linear subspace D(A) of the reflexive Banach space X. Then A generates a contraction semigroup if and only if Note that the conditions that D(A) is dense and that A is closed are dropped in comparison to the non-reflexive case. This is because in the reflexive case they follow from the other two conditions.

Dissipativity of the adjoint

Let A be a linear operator defined on a dense linear subspace D(A) of the reflexive Banach space X. Then A generates a contraction semigroup if and only if In case that X is not reflexive, then this condition for A to generate a contraction semigroup is still sufficient, but not necessary.

Quasicontraction semigroups

Let A be a linear operator defined on a linear subspace D(A) of the Banach space X. Then A generates a quasi contraction semigroup if and only if

Examples

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. The following is an example in point.