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.

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,
2. A is closed,
3. A is dissipative, and
4. A − λ₀I is surjective for some λ₀> 0, where I denotes the identity operator.

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

1. A is dissipative, and
2. 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

1. D(A) is dense in X,
2. A is closed,
3. A is quasidissipative, i.e. there exists a ω≥0 such that ωI-A is dissipative operator, and
4. 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),

$\langle u,Au\rangle =\int _{0}^{1}u(x)u'(x)\,\mathrm {d} x=-{\frac {1}{2}}u(0)^{2}\leq 0,$

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

$u(x)={\rm {e}}^{\lambda x}\int _{x}^{1}{\rm {e}}^{-\lambda t}f(t)\,dt$

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.

References

• Lumer, Günter and Phillips, R. S. (1961). "Dissipative operators in a Banach space". Pacific J. Math. 11: 679–698. ISSN 0030-8730.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Renardy, Michael and 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.{{cite book}}: CS1 maint: multiple names: authors list (link)
• 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