  For positive $C_0$-semigroups $S$ and $T$ on a Banach lattice such that $S(t)
\le T(t)$ for all times $t$, we prove that analyticity of $T$ implies
analyticity of $S$. This answers an open problem posed by Arendt in 2004.
  Our proof is based on a spectral theoretic argument: we apply
Perron-Frobenius theory to multiplication operators that are induced by $S$ and
$T$ on a vector-valued function space.
