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  "body": "\n## Statement\n\nLet $H$ be a [[Hilbert space|HM2HHE3J51]] and let $T \\colon H \\to H$ be a contraction, that is, a bounded linear operator with $\\norm{T} \\le 1$.\nThen there is a Hilbert space $K$ that contains $H$ as a closed subspace.\nThere is also a [[unitary operator|MM0EJ2GDE4]] $U \\colon K \\to K$ such that\n$$T^n = P_H U^n|_H \\quad \\text{for all integers } n \\ge 0,$$\nwhere $P_H$ is the [[orthogonal projection|0B8KZRT28M]] of $K$ onto $H$.\nMoreover, $K$ can be chosen as the closed span of $\\bigcup_{n \\in \\Z} U^n H$.\nWith this choice, the pair $(K, U)$ is unique up to a unitary isomorphism that fixes $H$.\n\n## Remarks\n\nProof idea: the defect operator $D = (I - T^*T)^{1/2}$ satisfies $\\norm{Tx}^2 + \\norm{Dx}^2 = \\norm{x}^2$.\nOn $\\bigoplus_{n \\ge 0} H$, the operator $(x_0, x_1, \\ldots) \\mapsto (T x_0, D x_0, x_1, x_2, \\ldots)$ is an isometry, and it compresses to the powers of $T$ on the first copy of $H$.\nEvery isometry extends to a unitary operator on a larger space, which gives $U$.\n\nExample: the unilateral shift on $\\ell^2(\\N)$ is a contraction, and the bilateral shift on $\\ell^2(\\Z)$ is a unitary dilation of it.\n\nThe compression property holds only for powers $n \\ge 0$; for negative $n$ one has $T^{*\\abs{n}} = P_H U^n|_H$ instead.\n\n## Explanations\n\n### L1 human\n\nA contraction is an operator that never increases lengths.\nThe theorem says that every contraction is the shadow of a unitary operator, which preserves lengths, acting on a bigger space.\nTo see $T^n x$, apply $U$ to $x$ $n$ times in the big space and then project back onto $H$.\n\n### L2 human\n\nThe dilation turns questions about a contraction into questions about a unitary operator, where the spectral theorem is available.\nAn immediate consequence is [[von Neumann's inequality|7KS1W5K54S]].\nThe minimal unitary dilation is the starting point of the Sz.-Nagy–Foias model theory of contractions and of the [[commutant lifting theorem|3TVXB7X266]].\n",
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