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&nbsp;<&nbsp;θ&nbsp;<&nbsp;π/2, the continuous linear operator exp(At)&nbsp;:&nbsp;X&nbsp;→&nbsp;X can be extended to t&nbsp;∈&nbsp;Δ_θ,

:

and the usual semigroup conditions hold for s,&nbsp;t&nbsp;&isin;&nbsp;&Delta;_&theta;: exp(A0)&nbsp;=&nbsp;id, exp(A(t&nbsp;+&nbsp;s))&nbsp;=&nbsp;exp(At) exp(As), and, for each x&nbsp;&isin;&nbsp;X, exp(At)x is continuous in t;

• and, for all t&nbsp;∈&nbsp;Δ_θ&nbsp;\&nbsp;{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&nbsp;∈&nbsp;R such that the half-plane Re(λ)&nbsp;&gt;&nbsp;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)&nbsp;&gt;&nbsp;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&nbsp;&gt;&nbsp;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&nbsp;<&nbsp;θ&nbsp;<&nbsp;π/2&nbsp;+&nbsp;δ.

References