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  "body": "\n## Statement\n\nLet $X$ be a real or complex [[Banach space|06GZ956657]] and let $A$ be a [[densely defined|BCJHNHKQFG]] linear operator in $X$ with domain $D(A)$.\nThen the following are equivalent.\n\n1. $A$ is the [[generator|60HSWZBTK9]] of a [[contraction semigroup|XPWE94BV6N]] on $X$.\n2. $A$ is [[dissipative|3S7HJ5GQRF]], and there is $\\lambda_0 \u003e 0$ such that $\\lambda_0 I - A$ maps $D(A)$ onto $X$.\n\n## Remarks\n\nIf $A$ generates a contraction semigroup $T(t)$, then for $\\lambda \u003e 0$ the resolvent is the Laplace transform $(\\lambda I - A)^{-1}x = \\int_0^\\infty e^{-\\lambda t} T(t)x \\, dt$.\nThis gives surjectivity of $\\lambda I - A$ and the bound $\\norm{(\\lambda I - A)^{-1}} \\le 1/\\lambda$, which is dissipativity.\nConversely, dissipativity and one surjective $\\lambda_0 I - A$ give $\\norm{(\\lambda I - A)^{-1}} \\le 1/\\lambda$ for all $\\lambda \u003e 0$, and the [[Hille–Yosida theorem|5YR4GJKPCV]] then produces the semigroup.\n\nExample: on $L^2(0, 1)$, let $Af = f''$ with domain $H^2(0, 1) \\cap H^1_0(0, 1)$.\nIt is dissipative since $\\ip{Af}{f} = -\\int_0^1 \\abs{f'}^2 \\, dx$, and $I - A$ is onto because the boundary problem $f - f'' = g$, $f(0) = f(1) = 0$ has a solution for every $g \\in L^2(0, 1)$.\nSo $A$ generates the heat semigroup with Dirichlet boundary conditions, and it is a contraction semigroup.\n\nThe range condition cannot be dropped.\nThe operator $f \\mapsto f''$ on smooth functions with compact support in $(0, 1)$ is densely defined and dissipative, but it is not closed, so it is not a generator.\n\n## Explanations\n\n### L1 human\n\nThe theorem tells when the equation $u' = Au$ has solutions that never grow in norm.\nTwo things are needed.\nThe operator must lose energy, which is dissipativity.\nAnd the equation $\\lambda u - Au = g$ must be solvable for every right side $g$, which rules out an operator with too small a domain.\n\n### L2 human\n\nThe Hille–Yosida theorem asks for the resolvent bound $\\norm{(\\lambda - A)^{-1}} \\le 1/\\lambda$ for every $\\lambda \u003e 0$.\nThe Lumer–Phillips theorem replaces this bound by dissipativity, which is often a one-line integration by parts, plus solvability for one $\\lambda_0$.\nFor a second-order elliptic operator with Dirichlet conditions, the solvability usually comes from the Lax–Milgram theorem.\nFor a self-adjoint operator $A \\le 0$ on a Hilbert space, both conditions hold, and the semigroup is $e^{tA}$ given by the spectral theorem.\n",
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