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  "body": "\n## Statement\n\nLet $H$ be a [[Hilbert space|HM2HHE3J51]] and let $U \\colon H \\to H$ be a bounded linear operator with $\\norm{U} \\le 1$, for example a [[unitary operator|MM0EJ2GDE4]].\nLet $P$ be the [[orthogonal projection|0B8KZRT28M]] of $H$ onto the closed subspace $\\set{v \\in H : Uv = v}$.\nThen for every $v \\in H$,\n$$\\lim_{n \\to \\infty} \\norm{\\frac{1}{n} \\sum_{k=0}^{n-1} U^k v - P v} = 0.$$\n\n## Remarks\n\nProof idea: let $F = \\set{v : Uv = v}$ and $B = \\set{w - Uw : w \\in H}$.\nFor $\\norm{U} \\le 1$ one shows that $Uv = v$ if and only if $U^* v = v$, and this gives $H = F \\oplus \\overline{B}$, an [[orthogonal decomposition|0XJXTBCB2T]].\nOn $F$ the averages are constantly $v$.\nOn $B$ they telescope: $\\frac{1}{n} \\sum_{k=0}^{n-1} U^k (w - Uw) = \\frac{1}{n}(w - U^n w)$, of norm at most $2\\norm{w}/n$.\nSince all averaging operators have norm at most $1$, the convergence to $0$ extends from $B$ to its closure.\n\nMain application: let $T$ be a [[measure-preserving transformation|556WBDAHNP]] of a probability space $(X, \\mathcal{A}, \\mu)$, and let $U f = f \\circ T$ on [[$L^2(\\mu)$|SASP14MA1Z]].\nThen $U$ is an isometry, and the theorem says that the Birkhoff averages of every $f \\in L^2(\\mu)$ converge in $L^2$ to the projection of $f$ onto the $T$-invariant functions.\nIf $T$ is ergodic, the invariant functions are the constants, and the limit is the constant $\\int_X f \\, d\\mu$.\n\nExample: for $U = -\\id$ on $H = \\C$, the fixed space is $\\set{0}$.\nThe averages of $v$ equal $v/n$ for odd $n$ and $0$ for even $n$, so they tend to $0 = Pv$.\n\n## Explanations\n\n### L1 human\n\nApply a norm-non-increasing operator again and again to a vector and average the results.\nThe averages always settle down, and the limit is the part of the vector that the operator does not move.\nThe parts that move get averaged out.\n\n### L2 human\n\nThis is the Hilbert space form of the ergodic theorem, proved by von Neumann in 1932, and it gives convergence in mean rather than at almost every point.\nThrough the composition operator $f \\mapsto f \\circ T$, it turns questions about a measure-preserving map into questions about a linear operator, which is the starting point of the spectral theory of dynamical systems.\nBirkhoff's theorem gives almost-everywhere convergence and needs a different, harder argument based on a maximal inequality.\n",
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