---
tag: 2V1DXXM9FA
kind: statement
role: theorem
name: Beurling's theorem
msc: [47A15, 30H10, 30J05]
uses: [PNNNFJVF6W, NN8XH5RJYB, 0EP59ESQ94]
about: [PNNNFJVF6W, NN8XH5RJYB, 0EP59ESQ94]
sources:
  - url: https://en.wikipedia.org/wiki/Beurling–Lax_theorem
---

## Statement

Let $H^2$ be the [[Hardy space|PNNNFJVF6W]] of the unit disc $\mathbb{D}$, a [[Hilbert space|HM2HHE3J51]].
Let $S \colon H^2 \to H^2$ be the shift operator $(S f)(z) = z f(z)$.
Let $M \neq \set{0}$ be a closed subspace of $H^2$.
Then $M$ is an [[invariant subspace|0EP59ESQ94]] of $S$, that is $S M \subseteq M$, if and only if
$$M = \theta H^2 = \set{\theta f : f \in H^2}$$
for some [[inner function|NN8XH5RJYB]] $\theta$.
The inner function $\theta$ is unique up to a constant factor of modulus $1$.

## Remarks

Proof idea: $S$ is an isometry and $\bigcap_n S^n M = \set{0}$, so $M \ominus S M$ is not zero.
Take $\theta$ of norm $1$ in $M \ominus S M$.
Then $\theta \perp S^n \theta$ for $n \ge 1$, which says that all Fourier coefficients of $\abs{\theta^*}^2$ vanish except the one of index $0$; hence $\abs{\theta^*} = 1$ almost everywhere and $\theta$ is inner.
A decomposition of $M$ into the orthogonal pieces $S^n (M \ominus S M)$ then shows $M = \theta H^2$.

Example: the functions in $H^2$ that vanish at a point $a \in \mathbb{D}$ form a closed invariant subspace; it equals $\theta H^2$ with $\theta(z) = \frac{z - a}{1 - \bar a z}$.

A consequence: $f \in H^2$ is cyclic for $S$, meaning that the polynomial multiples of $f$ are dense, if and only if $f$ is an [[outer function|KGJD4A4ATY]].

## Explanations

### L1 human

Multiplying by $z$ is a basic operator on the space $H^2$.
The theorem lists every closed subspace that this operator maps into itself.
Each one consists of all multiples of a single inner function.
So a question about operators becomes a question about functions, which complex analysis can answer.

### L2 human

Beurling's theorem was among the first complete descriptions of the invariant subspaces of a natural operator on an infinite-dimensional space.
Together with the [[inner-outer factorization|X6ZQPAQF76]], it shows that the invariant subspace generated by $f$ is $I H^2$, where $I$ is the inner factor of $f$.
Lax extended it to vector-valued Hardy spaces, which is the Beurling–Lax theorem; the model theory of contractions builds on it.
Its dual form describes the invariant subspaces of the backward shift as the model spaces $H^2 \ominus \theta H^2$.