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.
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 ∈ Δθ,
- and the usual semigroup conditions hold for s, t ∈ Δθ: exp(A0) = id, exp(A(t + s)) = exp(At)exp(As), and, for each x ∈ X, exp(At)x is continuous in t;
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 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.