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  "body": "\n## Summary\n\nA composition operator acts on a space of functions by precomposing with a fixed map: $C_\\varphi f = f \\circ \\varphi$.\nIt is always linear.\nThe questions are when it maps the space into itself, when it is bounded or compact, and how its properties reflect the symbol $\\varphi$.\nOn the [[Hardy space|PNNNFJVF6W]] of the disc, every holomorphic self-map $\\varphi$ of the disc gives a bounded composition operator, by Littlewood's subordination principle.\nIn ergodic theory the same construction is called the Koopman operator.\n\n## Explanations\n\n### L1 human\n\nA composition operator changes a function by first moving the point with a fixed map $\\varphi$ and then applying the function.\nIf $\\varphi(z) = z/2$, then $C_\\varphi$ turns $f(z)$ into $f(z/2)$, a squeezed copy of $f$.\nOn power series $\\sum a_n z^n$ this multiplies the $n$-th coefficient by $2^{-n}$.\nProperties of the operator, such as its norm or its spectrum, then become questions about how $\\varphi$ moves points around.\n",
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