  We show that a bounded analytic semigroup on an $L_p$-space has a bounded
$H^{\infty}(\Sigma_{\varphi})$-calculus for some $\varphi < \frac{\pi}{2}$ if
and only if the semigroup can be obtained, after restricting to invariant
subspaces, factorizing through invariant subspaces and similarity transforms,
from a bounded analytic semigroup on some bigger $L_p$-space which is positive
and contractive on the real line.
