Analytic semigroup
# 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 < θ < π/ 2, the continuous linear operator exp(At) : X → X can be extended to t ∈ $Δ_θ$ ,

       $\Delta _{\theta }=\{0\}\cup \{t\in \mathbb {C} :|\mathrm {arg} (t)|<\theta \},$
    and the usual semigroup conditions hold for s, t ∈ $Δ_θ$ : exp(A0) = id, exp(A(t + s)) = exp(At) exp(As), and, for each x ∈ X, exp(At)x is continuous in t;
* and, for all t ∈ $Δ_θ$ \ {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 ∈ 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

    $\|R_{\lambda }(A)\|\leq {\frac {C}{|\lambda -\omega |}}$

for Re(λ) > ω and where $R_{\lambda }(A)$ 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

    $\left\{\lambda \in \mathbf {C} :|\mathrm {arg} (\lambda -\omega )|<{\frac {\pi }{2}}+\delta \right\}$

for some δ > 0, and an analogous resolvent estimate holds in this sector. Moreover, the semigroup is represented by

    $\exp(At)={\frac {1}{2\pi i}}\int _{\gamma }e^{\lambda t}(\lambda \mathrm {id} -A)^{-1}\,\mathrm {d} \lambda ,$

where γ is any curve from $e^{−iθ}∞$ to $e^{+iθ}∞$ such that γ lies entirely in the sector

    ${\big \\lambda \in \mathbf {C} :|\mathrm {arg} (\lambda -\omega )|\leq \theta {\big \}},$

with π/ 2 < θ < π/ 2 + δ.

