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  "body": "\n## Statement\n\nLet $X$ be a [[Banach space|06GZ956657]] and let $\\mathcal{L}(X)$ be the space of bounded linear operators on $X$ with the [[operator norm|06EKW1AKXF]].\nLet $T \\colon [0, \\infty) \\to \\mathcal{L}(X)$ satisfy:\n\n1. $T(0) = I$, the identity operator;\n2. $T(s + t) = T(s) T(t)$ for all $s, t \\ge 0$;\n3. $\\lim_{t \\to 0^+} \\norm{T(t) - I} = 0$.\n\nThen there is a unique $A \\in \\mathcal{L}(X)$ such that $T(t) = e^{tA}$ for all $t \\ge 0$, where $e^{tA}$ is the [[exponential|KX3QRA3CEP]] of $tA$.\nConversely, for every $A \\in \\mathcal{L}(X)$ the map $t \\mapsto e^{tA}$ satisfies conditions 1 to 3.\n\n## Remarks\n\nConditions 2 and 3 make $T$ continuous in operator norm on $[0, \\infty)$.\nLet $V(t) = \\int_0^t T(s) \\, ds$; since $V(t)/t \\to I$ in norm, $V(t_0)$ is invertible for a small $t_0 \u003e 0$.\nThe functional equation gives $(T(h) - I) V(t_0) = V(t_0 + h) - V(h) - V(t_0)$, and dividing by $h$ shows that $T$ is differentiable at $0$ with derivative $A = (T(t_0) - I) V(t_0)^{-1}$.\nThen $T'(t) = A T(t)$, and uniqueness for this linear equation gives $T(t) = e^{tA}$.\nThe operator $A$ is the [[generator|60HSWZBTK9]] of the semigroup.\nFor $X = \\R^n$ this says that the continuous solutions $F \\colon [0, \\infty) \\to \\R^{n \\times n}$ of $F(s + t) = F(s) F(t)$ with $F(0) = I$ are exactly $F(t) = e^{tA}$.\nCondition 3 cannot be weakened to strong continuity.\nThe left shifts $(T(t)f)(s) = f(s + t)$ on $L^2(\\R)$ form a [[$C_0$-semigroup|EVF180QJXY]] with the unbounded generator $f \\mapsto f'$, so they are not of the form $e^{tA}$ with $A$ bounded.\nCondition 1 is needed: for a projection $P \\ne I$, the constant map $T(t) = P$ satisfies condition 2 and is norm continuous, but $T(0) = P \\ne e^{0A}$.\n\n## Explanations\n\n### L1 human\n\nThe only continuous real functions with $f(s + t) = f(s) f(t)$ and $f(0) = 1$ are $f(t) = e^{at}$.\nThis theorem is the same fact for operators: a family of operators that composes like exponentials, and moves continuously in norm, is an exponential $e^{tA}$.\nThe operator $A$ is the rate of change of the family at time $0$.\n\n### L2 human\n\nThe theorem identifies the norm-continuous one-parameter semigroups with the bounded operators $A$.\nMany evolution equations, such as the heat equation, give semigroups that are only strongly continuous, and then the generator is an unbounded operator.\nFor those semigroups the Hille–Yosida theorem takes the place of this result.\n",
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