Multiplication semigroup
= Multiplication semigroup
{title2=$P_tf=e^{-tk}f$}
A nonnegative continuous multiplier $k$ defines a <contraction semigroup> on the <space of continuous functions vanishing at infinity> and on $L^2(\mu)$ by $P_tf=e^{-tk}f$. <Strong continuity> follows from <uniform convergence> on <compact sets> and small tails in $C_0$, or the <dominated convergence theorem> in $L^2$. The multiplier itself may be unbounded.