Generator of a multiplication semigroup ($Lf=-kf,D(L)=\{f:kf\text{ belongs to the underlying space}\}$)

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.