# 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.

Contents 1 Statement of the theorem 2 Variants of the theorem 2.1 Reflexive spaces 2.2 Dissipativity of the adjoint 2.3 Quasicontraction semigroups 3 Examples 4 Notes 5 References 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[1] 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[2] 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[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(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), {displaystyle langle u,Aurangle =int _{0}^{1}u(x)u'(x),mathrm {d} x=-{frac {1}{2}}u(0)^{2}leq 0,} so that A is dissipative. The ordinary differential equation u' − λu = f, u(1) = 0 has a unique solution u in H1([0, 1]; R) for any f in L2([0, 1]; R), namely {displaystyle u(x)={rm {e}}^{lambda x}int _{1}^{x}{rm {e}}^{-lambda t}f(t),dt} so that the surjectivity condition is satisfied. Hence, 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. Theorem 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 (e.g. 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". Pacific J. Math. 11: 679–698. doi:10.2140/pjm.1961.11.679. ISSN 0030-8730. Renardy, Michael & Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second 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, Charles; Hieber, Matthias; Neubrander, Frank (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 (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Semigroup theoryTheorems in functional analysis

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