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  "body": "\n## Summary\n\nA contraction semigroup is a [[strongly continuous semigroup|EVF180QJXY]] $(T(t))_{t \\ge 0}$ of operators on a Banach space with $\\norm{T(t)} \\le 1$ for every $t \\ge 0$.\nIt describes a linear evolution that never increases the norm of the state.\nThe [[Lumer–Phillips theorem|Z2N8MPD8DH]] says that its generators are exactly the densely defined m-dissipative operators.\nHeat flows, diffusions and unitary groups restricted to $t \\ge 0$ give contraction semigroups.\n\n## Explanations\n\n### L1 human\n\nA semigroup moves a state forward in time: $T(t)x$ is the state at time $t$ when the start is $x$.\nFor a contraction semigroup, the size of the state can only shrink or stay the same.\nThe heat semigroup is an example: the total squared temperature of a rod with ends kept at zero goes down with time.\nOn the real line, $T(t)x = e^{-t}x$ is a contraction semigroup, while $T(t)x = e^{t}x$ is not.\nEvery strongly continuous semigroup with $\\norm{T(t)} \\le M e^{\\omega t}$ becomes a contraction semigroup after rescaling and a change of norm, which is why contractions are a central case.\n",
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