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 D(A) is dense in X, A is closed, A is dissipative, and A − λ0I is surjective for some λ0> 0, where I denotes the identity operator. 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 A is dissipative, and A − λ0I is surjective for some λ0> 0, where I denotes the identity operator. 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 A is closed and both A and its adjoint operator A∗ are dissipative. 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 D(A) is dense in X, A is closed, A is quasidissipative, i.e. there exists a ω ≥ 0 such that A − ωI is dissipative, and A − λ0I is surjective for some λ0 > ω, where I denotes the identity operator. Examples Consider H = L2([0, 1]; R) with its usual inner product, and let Au = u′ with domain D(A) equal to those functions u in the Sobolev space H1([0, 1]; R) with u(1) = 0. D(A) is dense. Moreover, for every u in D(A), so that A is dissipative. The ordinary differential equation u' − λu = f, u(1)