Kalpana Kalpana (Editor)

Analytic semigroup

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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.

Contents

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 ∈ Δθ,
    • 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;
  • 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) > ω is contained in the resolvent set of A and, moreover, there is a constant C such that

    ∥ R λ ( A ) ∥ ≤ C | λ − ω |

    for Re(λ) > ω and where R λ ( 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

    { λ ∈ C : | a r g ( λ − ω ) | < π 2 + δ }

    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 ,

    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 } ,

    with π ⁄ 2 < θ < π ⁄ 2 + δ.

    References

    Analytic semigroup Wikipedia
    (Text) CC BY-SA