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,

Δ θ = { 0 } ∪ { t ∈ C : | a r g ( t ) | < θ } , {\displaystyle \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 for the resolvent R λ ( A ) {\displaystyle R_{\lambda }(A)} of the operator A we have

‖ R λ ( A ) ‖ ≤ C | λ − ω | {\displaystyle \|R_{\lambda }(A)\|\leq {\frac {C}{|\lambda -\omega |}}}

for Re(λ) > ω. Such operators are called sectorial. If this is the case, then the resolvent set actually contains a sector of the form

{ λ ∈ C : | a r g ( λ − ω ) | < π 2 + δ } {\displaystyle \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 ⁡ ( A t ) = 1 2 π i ∫ γ e λ t ( λ i d − A ) − 1 d λ , {\displaystyle \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

{ λ ∈ C : | a r g ( λ − ω ) | ≤ θ } , {\displaystyle {\big \{}\lambda \in \mathbf {C} :|\mathrm {arg} (\lambda -\omega )|\leq \theta {\big \}},}

with π/2 < θ < π/2 + δ.

References

• Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. xiv+434. ISBN 0-387-00444-0. MR 2028503.