Analytic semigroup

In mathematics, an analytic semigroup is particular kind of strongly continuous semigroup. Analytic semigroups are used in the solution of partial differential equations; compared to strongly continuous semigroups, analytic semigroups provide better regularity of solutions to initial value problems, better results concerning perturbations of the infinitesimal generator, and a relationship between the type of the semigroup and the spectrum of the infinitesimal generator.

Definition

Let Γ("t") = exp("At") be a strongly continuous one-parameter semigroup on a Banach space ("X", ||·||) with infinitesimal generator "A". Γ is said to be an analytic semigroup if

* for some 0 < "θ" < "π" ⁄ 2, the continuous linear operator exp("At") : "X" → "X" can be extended to "t" ∈ Δ_"θ",

::

:and the usual semigroup conditions hold for "s", "t" ∈ Δ_"θ": exp("A"0) = id, exp("A"("t" + "s")) = exp("At")exp("As"), and, for each "x" ∈ "X", exp("At")"x" is continuous in "t";

* and, for all "t" ∈ Δ_"θ" {0}, exp("At") is analytic in "t" in the sense of the uniform operator topology.

Characterization

The infinitesimal generators of analytic semigroups have the following characterization:

A closed, densely-defined linear operator "A" on a Banach space "X" is the generator of an analytic semigroup if and only if there exists an "ω" ∈ R such that the half-plane Re("λ") > "ω" is contained in the resolvent set of "A" and, moreover, there is a constant "C" such that

:$|\; R\_\{lambda\}\; (A)\; |\; leq\; frac\{C\}$

for Re("λ") > "ω". If this is the case, then the resolvent set actually contains a sector of the form

:

for some "δ" > 0, and an analogous resolvent estimate holds in this sector. Moreover, the semigroup is represented by

:$exp\; (At)\; =\; frac1\{2\; pi\; i\}\; int\_\{gamma\}\; e^\{lambda\; t\}\; (\; lambda\; mathrm\{id\}\; -\; A\; )^\{-1\}\; ,\; mathrm\{d\}\; lambda,$

where "γ" is any curve from "e"^−"iθ"∞ to "e"^+"iθ"∞ such that "γ" lies entirely in the sector

:

with "π" ⁄ 2 < "θ" < "π" ⁄ 2 + "δ".

References

* cite book
last = Renardy
first = Michael
coauthors = 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 = pp. xiv+434
isbn = 0-387-00444-0 MathSciNet|id=2028503