On the structure of semigroups on $L_p$ with a bounded $H{^\infty}$-calculus

We show that a bounded analytic semigroup on an $L_p$-space has a bounded $H^{\infty}(Σ_φ)$-calculus for some $φ< \fracπ{2}$ 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.

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