Multiplication semigroup ($P_{t}f=e^{-tk}f$)

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

The infinitesimal generator of a semigroup for $P_{t}f=e^{-tk}f$ is multiplication by $-k$ on its maximal domain. The bound $|(1-e^{-tk})/t|\le k$ proves the norm derivative when $kf$ is integrable; in $C_0$ use a compact-small-tail argument on $kf$. Conversely norm convergence identifies the pointwise derivative as $-kf$. Limits of graph pairs satisfy the same multiplication identity, proving it is a closed operator.