---
tag: DZF985Y137
kind: definition
name: Unitary representation
msc: ["22D10", "43A65"]
defines: 74BPYJNZTM
uses: [92700QAB5Y, HM2HHE3J51, MM0EJ2GDE4, KHJ2G2R5G7, WYGRXCG2RN]
sources:
  - url: https://en.wikipedia.org/wiki/Unitary_representation
---

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] and let $H$ be a complex [[Hilbert space|HM2HHE3J51]].
Let $U(H)$ be the group of [[unitary operators|MM0EJ2GDE4]] on $H$.
A *unitary representation* of $G$ on $H$ is a map $\pi \colon G \to U(H)$ such that:

1. $\pi$ is a group homomorphism: $\pi(gh) = \pi(g)\pi(h)$ for all $g, h \in G$;
2. $\pi$ is strongly continuous: for every $v \in H$, the map $G \to H$, $g \mapsto \pi(g)v$, is continuous.

## Remarks

Clause 2 says that $\pi$ is continuous for the [[strong operator topology|KHJ2G2R5G7]] on $U(H)$.
It is equivalent to weak continuity, the continuity of $g \mapsto \ip{\pi(g)u}{v}$ for all $u, v \in H$.

Example: $G = (\R, +)$ on $H = L^2(\R)$ by $(\pi(t)f)(x) = f(x - t)$.
Translation preserves $\int \abs{f}^2$ and has inverse $\pi(-t)$, so each $\pi(t)$ is unitary, and $\pi(s + t) = \pi(s)\pi(t)$.
For continuity, $\norm{\pi(t)f - f}_2 \to 0$ as $t \to 0$ holds for compactly supported continuous $f$ by uniform continuity, and then for all $f \in L^2(\R)$ since these are dense and each $\pi(t)$ has norm $1$.

Non-example: let $\ell^2(\R)$ be the Hilbert space of square-summable functions on the set $\R$, with orthonormal basis $(\delta_s)_{s \in \R}$, and let $\pi(t)\delta_s = \delta_{s + t}$.
Each $\pi(t)$ is unitary and $\pi$ is a homomorphism, but clause 2 fails for the usual topology of $\R$.
Indeed $\norm{\pi(t)\delta_0 - \delta_0} = \sqrt{2}$ for every $t \ne 0$, so $\pi(t)\delta_0$ does not tend to $\delta_0$ as $t \to 0$.

Two unitary representations $\pi_1$ on $H_1$ and $\pi_2$ on $H_2$ are *unitarily equivalent* if there is a unitary $T \colon H_1 \to H_2$ with $T\pi_1(g) = \pi_2(g)T$ for all $g \in G$.
A unitary representation is *irreducible* if $H \ne 0$ and the only closed subspaces invariant under all $\pi(g)$ are $0$ and $H$.