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  "body": "\n## Definition\n\nLet $(X, \\mathcal{B}, \\mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \\colon X \\to X$ be a [[measure-preserving transformation|556WBDAHNP]].\nThe *Koopman operator* of $T$ is the [[composition operator|CFASPSF7BB]]\n$$U_T \\colon L^2(\\mu) \\to L^2(\\mu), \\qquad U_T f = f \\circ T,$$\non the [[space|SASP14MA1Z]] $L^2(\\mu)$ of square-integrable functions modulo equality almost everywhere.\n\n## Remarks\n\nThe operator is well defined and isometric: since $\\mu(T^{-1}A) = \\mu(A)$ for all $A \\in \\mathcal{B}$, one gets $\\int \\abs{f \\circ T}^2 \\, d\\mu = \\int \\abs{f}^2 \\, d\\mu$.\nIf $T$ is invertible with measurable measure-preserving inverse, then $U_T$ is a [[unitary operator|MM0EJ2GDE4]] with inverse $U_{T^{-1}}$.\n\nExample: for the rotation $T(x) = x + \\alpha \\bmod 1$ with Lebesgue measure, the functions $e_k(x) = e^{2\\pi i k x}$ satisfy $U_T e_k = e^{2\\pi i k \\alpha} e_k$.\nThey form an orthonormal basis of $L^2$, so $U_T$ is diagonal in this basis.\n\nNon-example: if $T$ does not preserve $\\mu$, the formula need not define a bounded operator on $L^2(\\mu)$.\nFor $T(x) = x^2$ on $[0,1]$ with Lebesgue measure and $f(x) = x^{-1/3}$, the function $f$ is square-integrable but $f \\circ T = x^{-2/3}$ is not.\n\nThe same formula defines an isometry of $L^p(\\mu)$ for every $1 \\le p \\le \\infty$.\nThe constants are always fixed by $U_T$, and $T$ is ergodic if and only if they are the only fixed vectors.\n",
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