Teorema di 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.

Contenuti 1 Enunciato del teorema 2 Variants of the theorem 2.1 Reflexive spaces 2.2 Dissipativity of the adjoint 2.3 Quasicontraction semigroups 3 Esempi 4 Appunti 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] D(UN) is dense in X, A is closed, A is dissipative, and A − λ0I is surjective for some λ0> 0, dove I denota l'operatore di identità.

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, dove I denota l'operatore di identità.

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, cioè. there exists a ω ≥ 0 such that A − ωI is dissipative, and A − λ0I is surjective for some λ0 > ω, dove I denota l'operatore di identità. 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. D(UN) è denso. Inoltre, for every u in D(UN), {displaystyle langle u,Aurangle =int _{0}^{1}tu(X)u'(X),matematica {d} 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), vale a dire {stile di visualizzazione u(X)={rm {e}}^{lambda x}int _{1}^{X}{rm {e}}^{-lambda t}f(t),dt} so that the surjectivity condition is satisfied. Quindi, 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. Teorema 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 (per esempio. 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". Pacifico J. Matematica. 11: 679–698. doi:10.2140/pjm.1961.11.679. ISSN 0030-8730. Renardo, Michael & Rogers, Robert C. (2004). Introduzione alle equazioni alle derivate parziali. Testi in Matematica Applicata 13 (Seconda ed.). New York: Springer-Verlag. p. 356. ISBN 0-387-00444-0. Engel, Klaus-Jochen; Nagel, Rainer (2000), One-parameter semigroups for linear evolution equations, Springer Arendt, Wolfgang; Batty, Carlo; Hieber, Mattia; Neubrander, Franco (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 (argomenti – glossario) Spazi BanachBesovFréchetHilbertHölderNucleareOrliczSchwartzSobolevVettore topologico Proprietà barrelledcompletatodual (algebrico/topologico)localmente convessoriflessivoseparabile TeoremiHahn–BanachRieszrappresentazionegrafo chiusoprincipio di limitatezza uniformeKakutani punto fissoKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operatori adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebres Algebra di BanachC*-algebraspettro di un'algebra C*problemi di un operatore algebra localmente compatto di un'algebra di Neumanngruppo compatto di un'algebra di Neumann Problema del sottospazio Congettura di Mahler Applicazioni Spazio di Hardy Teoria spettrale delle equazioni differenziali ordinarie Heat Kernel Teorema dell'indice Calcolo delle variazioni Calcolo funzionale Operatore integrale Polinomio di Jones Teoria dei campi quantistici topologici Geometria non commutativa Ipotesi di Riemann Distribuzione (o funzioni generalizzate) Argomenti avanzati proprietà di approssimazione insieme bilanciato Teoria di Choquet topologia debole Distanza di Banach–Mazur Teoria di Tomita–Takesaki Categorie: Semigroup theoryTheorems in functional analysis