Lumer–Phillips theorem

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.

Contenu 1 Énoncé du théorème 2 Variants of the theorem 2.1 Reflexive spaces 2.2 Dissipativity of the adjoint 2.3 Quasicontraction semigroups 3 Exemples 4 Remarques 5 References Statement of the theorem Let A be a linear operator defined on a linear subspace D(UN) of the Banach space X. Then A generates a contraction semigroup if and only if[1] ré(UN) is dense in X, A is closed, A is dissipative, and A − λ0I is surjective for some λ0> 0, où je désigne l'opérateur d'identité.

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(UN) of the reflexive Banach space X. Then A generates a contraction semigroup if and only if[2] A is dissipative, and A − λ0I is surjective for some λ0> 0, où je désigne l'opérateur d'identité.

Note that the conditions that D(UN) 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(UN) of the reflexive Banach space X. Then A generates a contraction semigroup if and only if[3] 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.[4] Quasicontraction semigroups Let A be a linear operator defined on a linear subspace D(UN) of the Banach space X. Then A generates a quasi contraction semigroup if and only if D(UN) is dense in X, A is closed, A is quasidissipative, c'est à dire. there exists a ω ≥ 0 such that A − ωI is dissipative, and A − λ0I is surjective for some λ0 > ω, où je désigne l'opérateur d'identité. Examples Consider H = L2([0, 1]; R) with its usual inner product, and let Au = u′ with domain D(UN) equal to those functions u in the Sobolev space H1([0, 1]; R) with u(1) = 0. ré(UN) is dense. En outre, for every u in D(UN), {displaystyle langle u,Aurangle =int _{0}^{1}tu(X)u'(X),mathrm {ré} x=-{frac {1}{2}}tu(0)^{2}leq 0,} so that A is dissipative. The ordinary differential equation u' − λu = f, tu(1) = 0 has a unique solution u in H1([0, 1]; R) for any f in L2([0, 1]; R), à savoir {style d'affichage u(X)={rm {e}}^{lambda x}entier _{1}^{X}{rm {e}}^{-lambda t}F(t),dt} so that the surjectivity condition is satisfied. Ainsi, 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.

A normal operator (an operator that commutes with its adjoint) on a Hilbert space generates a strongly continuous semigroup if and only if its spectrum is bounded from above.[5] Notes ^ Engel and Nagel Theorem II.3.15, Arendt et al. Théorème 3.4.5, Staffans Theorem 3.4.8 ^ Engel and Nagel Corollary II.3.20 ^ Engel and Nagel Theorem II.3.17, Staffans Theorem 3.4.8 ^ There do appear statements in the literature that claim equivalence also in the non-reflexive case (par exemple. Luo, Guo, Morgul Corollary 2.28), but these are in error. ^ Engel and Nagel Exercise II.3.25 (ii) References Lumer, Günter & Phillips, R. S. (1961). "Dissipative operators in a Banach space". Pacifique J. Math. 11: 679–698. est ce que je:10.2140/pjm.1961.11.679. ISSN 0030-8730. Renardy, Michael & Rogers, Robert C.. (2004). Une introduction aux équations aux dérivées partielles. Textes en mathématiques appliquées 13 (Deuxième éd.). New York: Springer Verlag. p. 356. ISBN 0-387-00444-0. Engel, Klaus-Jochen; Nagel, Pluie (2000), One-parameter semigroups for linear evolution equations, Springer Arendt, Wolfgang; Batty, Charles; Hieber, Mathias; Neubrander, Franc (2001), Vector-valued Laplace Transforms and Cauchy Problems, Birkhauser Staffans, Olof (2005), Well-posed linear systems, Cambridge University Press Luo, Zheng-Hua; Guo, Bao-Zhu; Morgul, Omer (1999), Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer hide vte Functional analysis (sujets – glossaire) Espaces BanachBesovFréchetHilbertHölderNucléaireOrliczSchwartzSobolevvecteur topologique Propriétés tonneaucomplètedouble (algébrique/topologique)localement convexe réflexif séparable Théorèmes Hahn–Banach Représentation de Riesz graphe fermé principe de délimitation uniforme Kakutani virgule fixeKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Opérateurs adjointlimitécompactHilbert–Schmidtnormalnucléairetraceclasstransposéillimitéunitaire problème de sous-espaceconjecture de MahlerApplicationsespace de Hardythéorie spectrale des équations différentielles ordinairesnoyau de chaleurthéorème d'indexcalcul des variationscalcul fonctionnelopérateur intégralpolynôme de Jonesthéorie des champs quantiques topologiquesgéométrie non commutativehypothèse de Riemanndistribution (ou fonctions généralisées) Sujets avancés propriété d'approximationensemble équilibréThéorie de Choquettopologie faibleDistance de Banach–MazurThéorie de Tomita–Takesaki Catégories: Semigroup theoryTheorems in functional analysis