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.

tatement 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" − "λ"₀"I" is surjective for some "λ"₀ > "ω", where "I" denotes the identity operator. Then "A" generates a quasicontraction semigroup and

:

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" = "L"²( [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 "H"¹( [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"),

::$langle\; u,\; A\; u\; angle\; =\; int\_\{0\}^\{1\}\; u(x)\; u\text{'}(x)\; ,\; mathrm\{d\}\; x\; =\; -\; frac1\{2\}\; u(0)^\{2\}\; leq\; 0.$

: Hence, "A" generates a contraction semigroup.

References

* cite journal
author = Lumer, Günter and Phillips, R. S.
title = Dissipative operators in a Banach space
journal = Pacific J. Math.
volume = 11
year = 1961
pages = 679–698
issn = 0030-8730
* cite book
author = Renardy, Michael and Rogers, Robert C.
title = An introduction to partial differential equations
series = Texts in Applied Mathematics 13
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 2004
pages = 356
id = ISBN 0-387-00444-0 (Theorem 11.22)