# Teorema de Lumer-Phillips 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.

Conteúdo 1 Declaração do teorema 2 Variants of the theorem 2.1 Reflexive spaces 2.2 Dissipativity of the adjoint 2.3 Quasicontraction semigroups 3 Exemplos 4 Notas 5 References Statement of the theorem Let A be a linear operator defined on a linear subspace D(UMA) of the Banach space X. Then A generates a contraction semigroup if and only if D(UMA) is dense in X, A is closed, A is dissipative, and A − λ0I is surjective for some λ0> 0, onde I denota o operador identidade.

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(UMA) 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, onde I denota o operador identidade.

Note that the conditions that D(UMA) 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(UMA) 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(UMA) of the Banach space X. Then A generates a quasi contraction semigroup if and only if D(UMA) is dense in X, A is closed, A is quasidissipative, ou seja. there exists a ω ≥ 0 such that A − ωI is dissipative, and A − λ0I is surjective for some λ0 > ω, onde I denota o operador identidade. Examples Consider H = L2([0, 1]; R) with its usual inner product, and let Au = u′ with domain D(UMA) equal to those functions u in the Sobolev space H1([0, 1]; R) with u(1) = 0. D(UMA) is dense. Além disso, for every u in D(UMA), {displaystyle langle u,Aurangle =int _{0}^{1}você(x)u'(x),matemática {d} x=-{fratura {1}{2}}você(0)^{2}leq 0,} so that A is dissipative. The ordinary differential equation u' − λu = f, você(1) = 0 has a unique solution u in H1([0, 1]; R) for any f in L2([0, 1]; R), nomeadamente {estilo de exibição você(x)={rm {e}}^{lambda x}int_{1}^{x}{rm {e}}^{-lambda t}f(t),dt} so that the surjectivity condition is satisfied. Por isso, by the reflexive version of the Lumer–Phillips theorem A generates a contraction semigroup.

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.