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  "body": "\n## Definition\n\nLet $X$ be a complex [[Banach space|06GZ956657]] with identity map $I$, and let $T \\colon X \\to X$ be a [[continuous linear map|03TM135G0N]] with [[spectrum|BMKATJ802A]] $\\sigma(T)$.\nThe *resolvent set* of $T$ is $\\rho(T) = \\C \\setminus \\sigma(T)$.\nThe *resolvent* of $T$ is the function\n$$R(\\cdot, T) \\colon \\rho(T) \\to \\mathcal{B}(X), \\qquad R(\\lambda, T) = (T - \\lambda I)^{-1},$$\nwhere $\\mathcal{B}(X)$ is the space of bounded operators on $X$ and the inverse is the [[bounded inverse|CPAD5W1CGW]].\n\n## Remarks\n\nSome authors use $(\\lambda I - T)^{-1}$ instead, which changes the sign.\nThe resolvent set is open, and $R(\\cdot, T)$ is a [[holomorphic function|0NAHXR1FRH]] with values in $\\mathcal{B}(X)$.\nIt satisfies the *resolvent identity*\n$$R(\\lambda, T) - R(\\mu, T) = (\\lambda - \\mu) R(\\lambda, T) R(\\mu, T) \\qquad (\\lambda, \\mu \\in \\rho(T)),$$\nand $\\norm{R(\\lambda, T)} \\ge 1 / \\operatorname{dist}(\\lambda, \\sigma(T))$.\n\nExample: for $D(x_n) = (x_n / n)$ on $\\ell^2$ and $\\lambda \\in \\rho(D)$, $R(\\lambda, D)$ is the diagonal operator with entries $1/(1/n - \\lambda)$.\n\nNon-example: $\\lambda = 0$ is not in $\\rho(D)$, since $D$ is not surjective.\nIndeed $(1/n)$ is in $\\ell^2$ but $(1)$ is not, so $(1/n)$ is not in the image of $D$.\nHence $R(0, D)$ is not defined.\n\nThe same definitions apply to closed operators, with $R(\\lambda, T) \\colon X \\to D(T)$.\n",
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