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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 $\\norm{T} \\le 1$.\nLet $V \\colon K \\to K$ be a minimal isometric dilation of $T$.\nThis means that $K$ is a Hilbert space containing $H$ as a closed subspace and $V$ is an [[isometry|02HPP5VCZH]].\nIt also means that $T^n = P_H V^n|_H$ for all $n \\ge 0$, and that $K$ is the closed span of $\\bigcup_{n \\ge 0} V^n H$.\nHere $P_H$ is the [[orthogonal projection|0B8KZRT28M]] of $K$ onto $H$.\nLet $A \\colon H \\to H$ be a bounded linear operator with $AT = TA$.\nThen there is a bounded linear operator $B \\colon K \\to K$ such that\n$$BV = VB, \\qquad \\norm{B} = \\norm{A}, \\qquad P_H B = A P_H.$$\n\n## Remarks\n\nThe condition $P_H B = A P_H$ says that $B^*$ maps $H$ into $H$ and $B^*|_H = A^*$; so $B$ lifts $A$ to the dilation space without increasing the norm.\nThe proof constructs $B$ step by step on the pieces of $K$, with a one-step extension lemma for operator matrices that keeps the norm.\nThe theorem is due to Sz.-Nagy and Foias, and Sarason proved an earlier special case.\n\nExample: take $T$ to be the compression of the shift $S$ on the [[Hardy space|PNNNFJVF6W]] $H^2$ to a model space $H^2 \\ominus \\theta H^2$, with $\\theta$ an [[inner function|NN8XH5RJYB]].\nThen the lifts of operators commuting with $T$ commute with $S$, so they are multiplications by bounded analytic functions.\nThis gives Sarason's theorem: each operator commuting with $T$ is the compression of multiplication by some $\\varphi \\in H^\\infty$ with the same norm.\n\n## Explanations\n\n### L1 human\n\nA contraction can be seen as a piece of an isometry acting on a larger space.\nThe theorem says that any operator that commutes with the contraction can be extended to the larger space so that it commutes with the isometry, with no loss in norm.\n\n### L2 human\n\nCommutant lifting gives a unified operator-theoretic approach to interpolation problems.\nThe Nevanlinna–Pick and Carathéodory–Fejér problems ask for a bounded analytic function on the disc with prescribed values or Taylor coefficients and least supremum norm.\nBoth reduce to lifting an operator that commutes with a compressed shift.\nThe theorem builds on [[Sz.-Nagy's dilation theorem|YB7Q3EK0ND]], which provides the minimal isometric dilation.\n",
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