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  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|06GZ956657]].\nA *contraction semigroup* on $X$ is a [[strongly continuous semigroup|EVF180QJXY]] $(T(t))_{t \\ge 0}$ of bounded operators on $X$ such that\n$$\\norm{T(t)} \\le 1 \\quad \\text{for all } t \\ge 0,$$\nwhere $\\norm{\\cdot}$ is the [[operator norm|06EKW1AKXF]].\n\n## Remarks\n\nExample: on $L^p(\\R)$ with $1 \\le p \u003c \\infty$, let $(T(t)f)(s) = f(s + t)$.\nEach $T(t)$ preserves the $L^p$ norm, so $\\norm{T(t)} = 1$, and $t \\mapsto T(t)f$ is continuous for every $f$, so this is a contraction semigroup.\n\nNon-example: on $\\C$, the semigroup $T(t)z = e^{t}z$ is strongly continuous, but $\\norm{T(t)} = e^{t} \u003e 1$ for $t \u003e 0$.\n\nThe [[generator|60HSWZBTK9]] of a contraction semigroup is [[dissipative|3S7HJ5GQRF]].\nSome authors write $C_0$ contraction semigroup to stress strong continuity.\n",
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