 Multiplication semigroup

A nonnegative continuous multiplier $k$ defines a contraction semigroup on the space of continuous functions vanishing at infinity and on $L^2(\mu )$ by $P_{t}f=e^{-tk}f$. 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.