---
tag: GE05N4B84Q
kind: definition
name: Analytic semigroup
msc: [47D06, 47D03]
defines: JSNNSNB9MJ
uses: [06GZ956657, EVF180QJXY, 0NAHXR1FRH]
sources:
  - url: https://en.wikipedia.org/wiki/Analytic_semigroup
---

## Definition

Let $X$ be a complex [[Banach space|06GZ956657]] and let $(T(t))_{t \ge 0}$ be a [[strongly continuous semigroup|EVF180QJXY]] 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:

1. $T(z_1 + z_2) = T(z_1)T(z_2)$ for all $z_1, z_2 \in \Delta_\theta$;
2. the map $z \mapsto T(z)$ is [[holomorphic|0NAHXR1FRH]] on $\Delta_\theta$ with values in the bounded operators;
3. $T(z)x \to x$ 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|60HSWZBTK9]] 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 $0 < \theta' < \theta$.