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/&hairsp;2, the continuous linear operator exp(At) : X → X can be extended to t ∈ Δθ&hairsp;,


 * $$\Delta_{\theta} = \{ 0 \} \cup \{ t \in \mathbb{C} : | \mathrm{arg}(t) | < \theta \},$$


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


 * and, for all t ∈ Δθ \ {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) &gt; ω is contained in the resolvent set of A and, moreover, there is a constant C such that for the resolvent $$R_\lambda(A)$$ of the operator A we have


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

for Re(λ) &gt; ω. 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 &gt; 0, and an analogous resolvent estimate holds in this sector. Moreover, the semigroup is represented by


 * $$\exp (At) = \frac1{2 \pi i} \int_{\gamma} e^{\lambda t} ( \lambda \mathrm{id} - A )^{-1} \, \mathrm{d} \lambda,$$

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/&hairsp;2 < θ < π/&hairsp;2 + δ.