Analytic semigroup

Builds on Banach space C0-semigroup Holomorphic function

Definition

Let $X$ be a complex Banach space and let $(T(t))_{t \ge 0}$ be a strongly continuous semigroup of bounded operators on $X$ . For $0 < \theta \le \pi/2$ , let $\Delta_\theta = \set{z \in \C : z \ne 0, \ \abs{\arg z} < \theta}$ . The semigroup is analytic if for some $\theta \in (0, \pi/2]$ there is a family $(T(z))_{z \in \Delta_\theta}$ of bounded operators on $X$ with these properties. It agrees with the semigroup on $(0, \infty)$ , and:

$T(z_1 + z_2) = T(z_1)T(z_2)$ for all $z_1, z_2 \in \Delta_\theta$ ;
the map $z \mapsto T(z)$ is holomorphic on $\Delta_\theta$ with values in the bounded operators;
$T(z)x \to x$ as $z \to 0$ in $\Delta_\theta$ , for every $x \in X$ .

Remarks

Example: let $P \ge 0$ be a self-adjoint operator on a Hilbert space and $T(t) = e^{-tP}$ . By the spectral theorem, $T(z) = e^{-zP}$ is defined for $\operatorname{Re} z > 0$ , with $\norm{T(z)} \le 1$ . It satisfies the three clauses with $\theta = \pi/2$ , so $e^{-tP}$ is analytic; the heat semigroup is the case $P = -\Delta$ .

Non-example: on $L^2(\R)$ , let $(T(t)f)(s) = f(s + t)$ . For an analytic semigroup, $T(t)x$ lies in the domain of the generator for every $t > 0$ and every $x$ . Here this would mean that every shifted $L^2$ function lies in the Sobolev space $H^1(\R)$ , which is false, so no extension satisfying the clauses exists.

The semigroup is a bounded analytic semigroup if moreover $\norm{T(z)}$ is bounded on each smaller sector $\Delta_{\theta'}$ with $0 < \theta' < \theta$ .

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Tag	`GE05N4B84Q`
URI	`m2://GE05N4B84Q`
Kind	definition
Writing	W0 stub
MSC	47D06, 47D03
Defines	Analytic semigroup
Uses	Banach space, C0-semigroup, Holomorphic function
Changed	2 November 2026

Analytic semigroup

Definition

Remarks

Reading path

Further reading

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