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  "body": "\n## Statement\n\nLet $H^2$ be the [[Hardy space|PNNNFJVF6W]] of the unit disc $\\mathbb{D}$, a [[Hilbert space|HM2HHE3J51]].\nLet $S \\colon H^2 \\to H^2$ be the shift operator $(S f)(z) = z f(z)$.\nLet $M \\neq \\set{0}$ be a closed subspace of $H^2$.\nThen $M$ is an [[invariant subspace|0EP59ESQ94]] of $S$, that is $S M \\subseteq M$, if and only if\n$$M = \\theta H^2 = \\set{\\theta f : f \\in H^2}$$\nfor some [[inner function|NN8XH5RJYB]] $\\theta$.\nThe inner function $\\theta$ is unique up to a constant factor of modulus $1$.\n\n## Remarks\n\nProof idea: $S$ is an isometry and $\\bigcap_n S^n M = \\set{0}$, so $M \\ominus S M$ is not zero.\nTake $\\theta$ of norm $1$ in $M \\ominus S M$.\nThen $\\theta \\perp S^n \\theta$ 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.\nA decomposition of $M$ into the orthogonal pieces $S^n (M \\ominus S M)$ then shows $M = \\theta H^2$.\n\nExample: 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 $\\theta(z) = \\frac{z - a}{1 - \\bar a z}$.\n\nA 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]].\n\n## Explanations\n\n### L1 human\n\nMultiplying by $z$ is a basic operator on the space $H^2$.\nThe theorem lists every closed subspace that this operator maps into itself.\nEach one consists of all multiples of a single inner function.\nSo a question about operators becomes a question about functions, which complex analysis can answer.\n\n### L2 human\n\nBeurling's theorem was among the first complete descriptions of the invariant subspaces of a natural operator on an infinite-dimensional space.\nTogether 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 $f$.\nLax extended it to vector-valued Hardy spaces, which is the Beurling–Lax theorem; the model theory of contractions builds on it.\nIts dual form describes the invariant subspaces of the backward shift as the model spaces $H^2 \\ominus \\theta H^2$.\n",
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