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  "body": "\n## Summary\n\nA $C_0$-semigroup, or strongly continuous semigroup, is a family $T(t)$, $t \\ge 0$, of bounded linear operators on a [[Banach space|06GZ956657]] that behaves like $e^{tA}$.\nIt satisfies $T(0) = I$ and $T(t + s) = T(t) T(s)$, and $T(t) x \\to x$ as $t \\to 0$ for each vector $x$.\nIts generator $A$ is in general an unbounded operator, and $u(t) = T(t) u_0$ solves $u' = A u$.\n$C_0$-semigroups give the solutions of linear evolution equations such as the heat equation, transport equations and Markov processes.\n\n## Explanations\n\n### L1 human\n\nLet $T(t)$ be the map that takes a state now to the state after time $t$.\nRunning for $s$ and then for $t$ is the same as running for $t + s$, which is the semigroup law.\nFor the transport equation $u_t = u_s$ on the line, $T(t)$ shifts a profile: $(T(t) f)(s) = f(s + t)$.\nStrong continuity says that a short shift barely changes the profile, measured in the chosen norm.\nFor a matrix $A$, $T(t) = e^{tA}$ is the simplest example.\n",
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