# 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

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

## 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∞ to e+∞ 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.

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