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  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|06GZ956657]] and let $(T(t))_{t \\ge 0}$ be a [[strongly continuous semigroup|EVF180QJXY]] of bounded operators on $X$.\nThe *generator* of $(T(t))_{t \\ge 0}$ is the linear [[operator|KRJX5RA0PW]] $A$ with domain\n$$D(A) = \\set{x \\in X : \\lim_{t \\to 0^+} \\frac{T(t)x - x}{t} \\text{ exists in } X},$$\ngiven by\n$$Ax = \\lim_{t \\to 0^+} \\frac{T(t)x - x}{t}.$$\n\n## Remarks\n\nThe generator is closed and densely defined, and it determines the semigroup.\nFor $x \\in D(A)$, the map $u(t) = T(t)x$ is differentiable, stays in $D(A)$, and satisfies $u'(t) = Au(t)$.\nOne writes $T(t) = e^{tA}$.\n\nExample: let $B$ be a bounded operator on $X$ and $T(t) = e^{tB} = \\sum_{k \\ge 0} t^k B^k / k!$, the [[exponential of a bounded operator|KX3QRA3CEP]].\nThen $\\norm{(T(t)x - x)/t - Bx} \\le \\sum_{k \\ge 2} t^{k-1}\\norm{B}^k \\norm{x}/k! \\to 0$, so the generator is $B$ with $D(B) = X$.\n\nExample: on $X = C_0(\\R)$, the continuous functions that tend to $0$ at infinity with the sup norm, let $(T(t)f)(s) = f(s + t)$.\nIts generator is $Af = f'$ with domain the $C^1$ functions $f$ such that $f$ and $f'$ lie in $C_0(\\R)$.\n\nNon-example: the operator $f \\mapsto f'$ on smooth functions with compact support is not the generator of this shift semigroup.\nIt is a proper restriction of $A$: the limit in the definition exists for more functions, and the generator is defined on all of them.\n\nThe [[Hille–Yosida theorem|5YR4GJKPCV]] and the [[Lumer–Phillips theorem|Z2N8MPD8DH]] say which operators are generators.\n",
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