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  "body": "\n## Definition\n\nLet $X$ be a complex Banach space with identity map $I$, and let $T \\colon X \\to X$ be a bounded operator with [[spectrum|BMKATJ802A]] $\\sigma(T)$ and [[resolvent|J5YNNKSB9P]] $R(\\lambda, T) = (T - \\lambda I)^{-1}$.\nAssume that $\\sigma(T) = \\sigma_1 \\cup \\sigma_2$, where $\\sigma_1$ and $\\sigma_2$ are disjoint closed sets.\nLet $\\Gamma$ be a finite union of positively oriented closed rectifiable curves in $\\C \\setminus \\sigma(T)$.\nAssume that $\\Gamma$ has winding number $1$ around each point of $\\sigma_1$ and $0$ around each point of $\\sigma_2$.\nThe *Riesz projector* of $T$ for $\\sigma_1$ is the operator\n$$P_{\\sigma_1} = -\\frac{1}{2\\pi i} \\oint_\\Gamma R(\\lambda, T) \\, d\\lambda = \\frac{1}{2\\pi i} \\oint_\\Gamma (\\lambda I - T)^{-1} \\, d\\lambda.$$\n\n## Remarks\n\nThe integral does not depend on the choice of $\\Gamma$, by the Cauchy theorem for analytic operator-valued functions.\nThe operator $P = P_{\\sigma_1}$ is a [[projection|033DA1WE8Y]], $P^2 = P$, and it commutes with $T$.\nSo $X = PX \\oplus (I - P)X$ with both summands closed and invariant under $T$, and the restrictions of $T$ have spectra $\\sigma_1$ and $\\sigma_2$.\n\nExample: let $T = \\operatorname{diag}(1, 2)$ on $\\C^2$ and $\\sigma_1 = \\set{1}$, with $\\Gamma$ the circle $\\abs{\\lambda - 1} = \\tfrac12$.\nThen $(\\lambda I - T)^{-1} = \\operatorname{diag}(1/(\\lambda - 1), 1/(\\lambda - 2))$, and [[Cauchy's integral formula|0DG10TVYPN]] gives $P_{\\sigma_1} = \\operatorname{diag}(1, 0)$.\n\nNon-example: for the right shift on $\\ell^2$, the spectrum is the closed unit disc, which is connected.\nA subset such as the closed disc of radius $\\tfrac12$ is not separated from the rest of the spectrum, so it has no Riesz projector.\nOnly $\\sigma_1 = \\emptyset$ and $\\sigma_1 = \\sigma(T)$ are allowed, giving $0$ and $I$.\n",
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