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
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
for Re("λ") > "ω" and where formula_3 is the resolvent of the operator "A". Such operators are called "sectorial". 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
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".