# 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},

- and the usual semigroup conditions hold for
*s*,*t*∈ Δ_{θ}: exp(*A*0) = 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 Re(*λ*) > *ω* and where 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

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

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

with *π* ⁄ 2 < *θ* < *π* ⁄ 2 + *δ*.