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  "body": "\n## Statement\n\nLet $\\mathbb{T}$ be the unit circle, let $H^2 \\subseteq L^2(\\mathbb{T})$ be the Hardy space, and let $\\varphi \\colon \\mathbb{T} \\to \\C$ be a continuous function with no zeros.\nThen the [[Toeplitz operator|GQ0X3R3WM2]] $T_\\varphi$ on $H^2$ is a [[Fredholm operator|8HWGKWD6ZM]].\nIts [[index|2V9Q4KHY1T]] is\n$$\\operatorname{ind} T_\\varphi = -\\operatorname{wind}(\\varphi, 0).$$\nHere $\\operatorname{wind}(\\varphi, 0)$ is the [[winding number|0NQSV2EWDR]] of the closed curve $\\varphi$ around $0$.\n\n## Remarks\n\nIdea of the proof: for continuous $\\varphi$ and $\\psi$, the operator $T_\\varphi T_\\psi - T_{\\varphi\\psi}$ is compact.\nSo $T_{1/\\varphi}$ is an inverse of $T_\\varphi$ modulo compact operators, and $T_\\varphi$ is Fredholm by Atkinson's theorem.\nThe index is constant along homotopies of nonvanishing symbols, and $\\varphi$ is homotopic to $z^n$ with $n = \\operatorname{wind}(\\varphi, 0)$.\nFor $n \\ge 0$, $T_{z^n}$ is the $n$-th power of the shift, which is injective with cokernel of dimension $n$, so its index is $-n$; for $n \u003c 0$ one takes adjoints.\nThe hypothesis that $\\varphi$ has no zeros is necessary: for continuous $\\varphi$, the operator $T_\\varphi$ is Fredholm if and only if $\\varphi$ has no zeros on $\\mathbb{T}$.\n\n## Explanations\n\n### L1 human\n\nA Toeplitz operator multiplies by a function on the circle and then discards negative frequencies.\nWhen the function never vanishes, the operator is nearly invertible, and the failure is measured by an integer.\nThat integer is minus the number of times the function winds around the origin.\nFor the simplest case, multiplication by $z$ shifts the coefficients of a power series one step and loses exactly one dimension.\n\n### L2 human\n\nThe theorem is the one-dimensional model of the Atiyah–Singer index theorem: an analytic index equals a topological invariant of the symbol.\nIt is also a basic example of index theory for $C^*$-algebras, through the Toeplitz extension $0 \\to \\mathcal{K} \\to \\mathcal{T} \\to C(\\mathbb{T}) \\to 0$, whose index map $K_1(C(\\mathbb{T})) \\to K_0(\\mathcal{K}) = \\Z$ is the winding number up to a sign convention.\nHigher-dimensional versions for Toeplitz operators on strictly pseudoconvex domains are due to Boutet de Monvel.\n",
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