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  "body": "\n## Definition\n\nLet $a = (a_n)_{n \\ge 0}$ be a sequence of complex numbers.\nLet $\\ell^2 = \\ell^2(\\set{0, 1, 2, \\ldots})$ be the [[space of square-summable sequences|08P6T2WH6R]] with standard basis $(e_k)_{k \\ge 0}$.\nThe *Hankel operator* $\\Gamma_a \\colon \\ell^2 \\to \\ell^2$ is the [[bounded linear operator|03TM135G0N]] with the [[Hankel matrix|PQ1D8FQB3E]] $(a_{j+k})_{j, k \\ge 0}$, that is,\n$$\\ip{\\Gamma_a e_k}{e_j} = a_{j+k} \\quad \\text{for all } j, k \\ge 0,$$\nwhen such a bounded operator exists.\n\n## Remarks\n\nExample: the Hilbert matrix, with $a_n = 1/(n + 1)$, defines a bounded Hankel operator; Hilbert's inequality gives $\\norm{\\Gamma_a} = \\pi$.\n\nNon-example: the identity matrix is not a Hankel matrix.\nIts entries at $(1, 1)$ and $(0, 2)$ have the same index sum $2$, but they are $1$ and $0$.\n\nNehari's theorem: $\\Gamma_a$ is bounded if and only if there is an essentially bounded function $\\psi$ on the unit circle with $\\hat\\psi(n) = a_n$ for all $n \\ge 0$.\nThen $\\norm{\\Gamma_a}$ is the least $\\norm{\\psi}_\\infty$ over such $\\psi$, where $\\hat\\psi(n)$ is the $n$-th Fourier coefficient.\nKronecker's theorem: $\\Gamma_a$ has finite rank if and only if the power series $\\sum_{n \\ge 0} a_n z^n$ is a rational function.\nIn the function form, for a bounded symbol $\\varphi$ on the circle, the Hankel operator is $H_\\varphi f = (I - P)(\\varphi f)$ from $H^2$ to $L^2 \\ominus H^2$, with $P$ the orthogonal projection onto $H^2$.\n",
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