---
tag: T7JBM8472G
kind: definition
name: Generator of a semigroup
msc: [47D06, 47D03]
defines: 60HSWZBTK9
uses: [06GZ956657, EVF180QJXY, KRJX5RA0PW]
sources:
  - url: https://en.wikipedia.org/wiki/C0-semigroup
---

## Definition

Let $X$ be a [[Banach space|06GZ956657]] and let $(T(t))_{t \ge 0}$ be a [[strongly continuous semigroup|EVF180QJXY]] of bounded operators on $X$.
The *generator* of $(T(t))_{t \ge 0}$ is the linear [[operator|KRJX5RA0PW]] $A$ with domain
$$D(A) = \set{x \in X : \lim_{t \to 0^+} \frac{T(t)x - x}{t} \text{ exists in } X},$$
given by
$$Ax = \lim_{t \to 0^+} \frac{T(t)x - x}{t}.$$

## Remarks

The generator is closed and densely defined, and it determines the semigroup.
For $x \in D(A)$, the map $u(t) = T(t)x$ is differentiable, stays in $D(A)$, and satisfies $u'(t) = Au(t)$.
One writes $T(t) = e^{tA}$.

Example: let $B$ be a bounded operator on $X$ and $T(t) = e^{tB} = \sum_{k \ge 0} t^k B^k / k!$, the [[exponential of a bounded operator|KX3QRA3CEP]].
Then $\norm{(T(t)x - x)/t - Bx} \le \sum_{k \ge 2} t^{k-1}\norm{B}^k \norm{x}/k! \to 0$, so the generator is $B$ with $D(B) = X$.

Example: on $X = C_0(\R)$, the continuous functions that tend to $0$ at infinity with the sup norm, let $(T(t)f)(s) = f(s + t)$.
Its generator is $Af = f'$ with domain the $C^1$ functions $f$ such that $f$ and $f'$ lie in $C_0(\R)$.

Non-example: the operator $f \mapsto f'$ on smooth functions with compact support is not the generator of this shift semigroup.
It is a proper restriction of $A$: the limit in the definition exists for more functions, and the generator is defined on all of them.

The [[Hille–Yosida theorem|5YR4GJKPCV]] and the [[Lumer–Phillips theorem|Z2N8MPD8DH]] say which operators are generators.