Lumer-Phillips theorem

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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.

[edit] 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

$\big\| \exp (A t) \big\| \leq \exp (\omega t)$

for all t ≥ 0.

[edit] 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),

$\langle u, A u \rangle = \int_{0}^{1} u(x) u'(x) \, \mathrm{d} x = - \frac1{2} u(0)^{2} \leq 0.$

Hence, A generates a contraction semigroup.

[edit] References

Lumer, Günter and Phillips, R. S. (1961). "Dissipative operators in a Banach space". Pacific J. Math. 11: 679–698. ISSN 0030-8730.
Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations, Second edition, Texts in Applied Mathematics 13, New York: Springer-Verlag, 356. ISBN 0-387-00444-0. (Theorem 11.22)