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 = exp be a strongly continuous one-parameter semigroup on a Banach space with infinitesimal generator A. Γ is said to be an analytic semigroup if- for some 0 < θ < π/2, the continuous linear operator exp : X → X can be extended to t ∈ Δθ,
- and, for all t ∈ Δθ \ , exp is analytic in t in the sense of the uniform operator topology.
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 >; ω is contained in the resolvent set of A and, moreover, there is a constant C such that for the resolvent of the operator A we have
for Re > ω. 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.