Gate orbit

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,

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

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

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

for Re(λ) > ω and where $R_{\lambda }(A)$ is the resolvent of the operator A. If this is the case, then the resolvent set actually contains a sector of the form

$\{\lambda \in \mathbf{C}:|\mathrm{a}\mathrm{r}\mathrm{g}(\lambda -\omega )|<\frac{\pi }{2}+\delta \}$

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

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

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

$\{\lambda \in \mathbf{C}:|\mathrm{a}\mathrm{r}\mathrm{g}(\lambda -\omega )|\le \theta \},$

with π ⁄ 2 < θ < π ⁄ 2 + δ.

References

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