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  "body": "\n## Summary\n\nA dissipative operator is a linear operator $A$ on a Banach space such that $\\norm{(\\lambda - A)x} \\ge \\lambda \\norm{x}$ for every $\\lambda \u003e 0$.\nOn a Hilbert space this means $\\operatorname{Re}\\ip{Ax}{x} \\le 0$ for every $x$ in the domain.\nThe densely defined maximal dissipative operators are exactly the generators of semigroups that do not increase norms, by the [[Lumer–Phillips theorem|Z2N8MPD8DH]].\nThe name comes from physics: the evolution $u' = Au$ loses energy and never gains it.\n\n## Explanations\n\n### L1 human\n\nThink of $u' = Au$ as a law of motion and of $\\norm{u}^2$ as an energy.\nOn a Hilbert space, the derivative of $\\norm{u(t)}^2$ is $2 \\operatorname{Re}\\ip{Au}{u}$, so dissipativity says the energy can only go down.\nThe heat equation is the main example: heat spreads out and the total squared temperature decreases.\nIn one dimension, $\\ip{f''}{f} = -\\int \\abs{f'}^2 \\, dx \\le 0$ for smooth $f$ that vanish at the ends, so the second derivative is dissipative.\nThe operator $-A$ of a dissipative $A$ is called accretive.\n",
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