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  "body": "\n## Definition\n\nLet $X$ be a complex [[Banach space|06GZ956657]] and let $(T(t))_{t \\ge 0}$ be a [[strongly continuous semigroup|EVF180QJXY]] of bounded operators on $X$.\nFor $0 \u003c \\theta \\le \\pi/2$, let $\\Delta_\\theta = \\set{z \\in \\C : z \\ne 0, \\ \\abs{\\arg z} \u003c \\theta}$.\nThe semigroup is *analytic* if for some $\\theta \\in (0, \\pi/2]$ there is a family $(T(z))_{z \\in \\Delta_\\theta}$ of bounded operators on $X$ with these properties.\nIt agrees with the semigroup on $(0, \\infty)$, and:\n\n1. $T(z_1 + z_2) = T(z_1)T(z_2)$ for all $z_1, z_2 \\in \\Delta_\\theta$;\n2. the map $z \\mapsto T(z)$ is [[holomorphic|0NAHXR1FRH]] on $\\Delta_\\theta$ with values in the bounded operators;\n3. $T(z)x \\to x$ as $z \\to 0$ in $\\Delta_\\theta$, for every $x \\in X$.\n\n## Remarks\n\nExample: let $P \\ge 0$ be a self-adjoint operator on a Hilbert space and $T(t) = e^{-tP}$.\nBy the spectral theorem, $T(z) = e^{-zP}$ is defined for $\\operatorname{Re} z \u003e 0$, with $\\norm{T(z)} \\le 1$.\nIt satisfies the three clauses with $\\theta = \\pi/2$, so $e^{-tP}$ is analytic; the heat semigroup is the case $P = -\\Delta$.\n\nNon-example: on $L^2(\\R)$, let $(T(t)f)(s) = f(s + t)$.\nFor an analytic semigroup, $T(t)x$ lies in the domain of the [[generator|60HSWZBTK9]] for every $t \u003e 0$ and every $x$.\nHere this would mean that every shifted $L^2$ function lies in the Sobolev space $H^1(\\R)$, which is false, so no extension satisfying the clauses exists.\n\nThe semigroup is a *bounded analytic semigroup* if moreover $\\norm{T(z)}$ is bounded on each smaller sector $\\Delta_{\\theta'}$ with $0 \u003c \\theta' \u003c \\theta$.\n",
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