Lumer–Phillips theorem

In mathematics, the Lumer-Phillips theorem is a result in the theory of semigroups that gives a sufficient condition for a linear operator in a Hilbert space to generate a quasicontraction semigroup.

Statement of the theorem

Let (H, ⟨ , ⟩) be a real or complex Hilbert space. Let A be a linear operator defined on a dense linear subspace D(A) of H, taking values in H. Suppose also that A is quasidissipative, i.e., for some ω ≥ 0, Re⟨x, Ax⟩ ≤ ω⟨x, x⟩ for every x in D(A). Finally, suppose that A − λ₀I is surjective for some λ₀ > ω, where I denotes the identity operator. Then A generates a quasicontraction semigroup and

${\big \|}\exp(At){\big \|}\leq \exp(\omega t)$

for all t ≥ 0.

Examples

• Any self-adjoint operator (A = A^∗) whose spectrum is bounded above generates a quasicontraction semigroup.
• Any skew-adjoint operator (A = −A^∗) generates a quasicontraction semigroup.
• 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 and the spectrum of A is empty. Moreover, for every u in D(A),

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

Hence, A generates a contraction semigroup.

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) (Theorem 11.22)