On measuring unboundedness of the $H^{\mathrm{\infty }}$-calculus for generators of analytic semigroups

We investigate the boundedness of the $H^{\mathrm{\infty }}$-calculus by estimating the bound $b(\epsilon )$ of the mapping $H^{\mathrm{\infty }}\to \mathcal{B}(X)$: $f\mapsto f(A)T(\epsilon )$ for $\epsilon$ near zero. Here, $-A$ generates the analytic semigroup $T$ and $H^{\mathrm{\infty }}$ is the space of bounded analytic functions on a domain strictly containing the spectrum of $A$. We show that $b(\epsilon )=\mathcal{O}(\mathrm{\mid }\log \epsilon \mathrm{\mid })$ in general, whereas $b(\epsilon )=\mathcal{O}(1)$ for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analytic semigroups. We discuss the sharpness of our bounds and show that single square function estimates yield $b(\epsilon )=\mathcal{O}(\sqrt{\mathrm{\mid }\log \epsilon \mathrm{\mid }})$.