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  "body": "\n## Statement\n\nLet $\\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]].\nLet $n \\ge 1$, let $z_1, \\ldots, z_n \\in \\mathbb{D}$ be distinct, and let $w_1, \\ldots, w_n \\in \\mathbb{D}$.\nLet $P$ be the [[Pick matrix|TSPTN464CM]] with entries $P_{ij} = \\frac{1 - w_i \\overline{w_j}}{1 - z_i \\overline{z_j}}$.\n\n1. There is a [[holomorphic function|0DG0Z72CQD]] $f \\colon \\mathbb{D} \\to \\mathbb{D}$ with $f(z_i) = w_i$ for $i = 1, \\ldots, n$ if and only if $P$ is [[positive semidefinite|0880F9EQCP]].\n2. Let $P$ be positive semidefinite and singular, of rank $m$.\nThen such an $f$ is unique, and it is a constant of modulus $1$ times a [[Blaschke product|9N6R5ADJXT]] of degree $m$.\n\n## Remarks\n\nProof idea for necessity: if $\\abs{f} \\le 1$, the multiplication operator by $f$ on the [[Hardy space|PNNNFJVF6W]] $H^2$ has norm at most $1$.\nIts adjoint sends the reproducing kernel $k_{z_i}(z) = \\frac{1}{1 - \\overline{z_i} z}$ to $\\overline{w_i} k_{z_i}$, and the condition $\\norm{M_f^* g} \\le \\norm{g}$ on combinations of these kernels is exactly $P \\ge 0$.\nSufficiency can be proved by Pick's original induction, which reduces $n$ points to $n - 1$ points with a Möbius change of variables, or by the commutant lifting theorem.\n\nExample: for $n = 2$ with $z_1 = w_1 = 0$, $z_2 = a \\ne 0$ and $w_2 = b$, the matrix is positive semidefinite exactly when $\\abs{b} \\le \\abs{a}$.\nThis is the [[Schwarz lemma|0MB48XDBY3]]; when $\\abs{b} = \\abs{a}$, the matrix has rank $1$ and the unique solution is $f(z) = (b/a) z$.\n\n## Explanations\n\n### L1 human\n\nYou choose some points in the disc and where you want them to go.\nThe question is whether a holomorphic map of the disc into itself can do this.\nThe theorem gives a test: build the Pick matrix and check that it has no negative eigenvalue.\nWhen the test passes only just, the map is forced and is a finite Blaschke product.\n\n### L2 human\n\nThe theorem generalises the Schwarz–Pick lemma from one pair of points to any finite set.\nIts operator form says that multipliers of $H^2$ of norm at most $1$ are characterised by positivity of a kernel.\nThis viewpoint later led to the theory of complete Pick spaces.\nThe same method treats interpolation in the upper half-plane and boundary interpolation problems, and it has applications in control theory.\nFor infinitely many points the condition becomes positivity of an infinite Pick matrix, and the interpolating sequences are characterised by Carleson.\n",
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