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  "body": "\n## Statement\n\nLet $X$ be a real or complex [[Banach space|0BDTTM24DP]].\nLet $A \\colon D(A) \\to X$ be a linear operator with domain $D(A) \\subseteq X$.\nThen $A$ is the [[generator|60HSWZBTK9]] of a [[strongly continuous|EVF180QJXY]] [[contraction semigroup|XPWE94BV6N]] $(T(t))_{t \\ge 0}$ on $X$ if and only if the following two conditions hold.\n\n1. $A$ is a [[closed operator|VM7B38PA87]] and is [[densely defined|BCJHNHKQFG]].\n2. Every real $\\lambda \u003e 0$ lies in the [[resolvent set|J5YNNKSB9P]] of $A$, and $\\norm{(\\lambda I - A)^{-1}} \\le \\dfrac{1}{\\lambda}$.\n\n## Remarks\n\nThe necessity comes from the resolvent formula $(\\lambda I - A)^{-1} x = \\int_0^\\infty e^{-\\lambda t} T(t) x \\, dt$, whose norm is at most $\\norm{x} / \\lambda$ when $\\norm{T(t)} \\le 1$.\nFor the converse, one forms the Yosida approximations $A_\\lambda = \\lambda A (\\lambda I - A)^{-1} = \\lambda^2 (\\lambda I - A)^{-1} - \\lambda I$.\nThey are bounded operators, the exponentials $e^{t A_\\lambda}$ are contractions, and they converge strongly as $\\lambda \\to \\infty$ to a semigroup with generator $A$.\nThe general form concerns semigroups with $\\norm{T(t)} \\le M e^{\\omega t}$ for constants $M \\ge 1$ and $\\omega \\in \\R$.\nThe operator $A$ generates such a semigroup if and only if it is closed and densely defined, $(\\omega, \\infty)$ lies in its resolvent set, and $\\norm{(\\lambda I - A)^{-n}} \\le M (\\lambda - \\omega)^{-n}$ for all $\\lambda \u003e \\omega$ and $n \\ge 1$.\nExample: on $X = \\C$, the operator $A x = a x$ with $\\operatorname{Re} a \\le 0$ satisfies $\\abs{(\\lambda - a)^{-1}} \\le 1/\\lambda$ for $\\lambda \u003e 0$, and it generates the contractions $T(t) = e^{a t}$.\nIf $a \u003e 0$ is real, then $\\lambda = a$ is not in the resolvent set, and indeed $e^{a t} \u003e 1$ for $t \u003e 0$, so no contraction semigroup has this generator.\nThe [[Lumer–Phillips theorem|Z2N8MPD8DH]] replaces condition 2 by dissipativity and a range condition, which are often easier to check.\n\n## Explanations\n\n### L1 human\n\nMany evolution equations have the form $u'(t) = A u(t)$, where $A$ is an unbounded operator such as a second derivative.\nOne wants a solution map $T(t)$ that sends the initial state to the state at time $t$.\nThe theorem says exactly which operators $A$ give such a family of maps that never increases the norm.\nThe test only uses the inverse of $\\lambda I - A$, which is often easy to study.\n\n### L2 human\n\nThe theorem turns an abstract Cauchy problem $u' = A u$, $u(0) = u_0$, in a [[Banach space|0BDTTM24DP]] into a question about the resolvent of $A$.\nFor a [[densely defined|BCJHNHKQFG]] closed operator, the resolvent bound is checked by solving the stationary equation $\\lambda u - A u = f$ with an a priori estimate $\\lambda \\norm{u} \\le \\norm{f}$.\nThis is how one shows that the heat equation and many parabolic equations are well posed.\nThe semigroup is then the abstract replacement for the matrix exponential $e^{tA}$ of a system of linear ordinary differential equations.\nThe [[Lumer–Phillips theorem|Z2N8MPD8DH]] is the form used most often in Hilbert spaces.\n",
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