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 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 the resolvent
of the operator A we have 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 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.