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

Abstract

We investigate the boundedness of the $H^{\infty }$-calculus by estimating the bound $b(\epsilon )$ of the mapping $H^{\infty }\to \mathcal{B}(X)$: $f\mapsto f(A)T(\epsilon )$ for $\epsilon$ near zero. Here, $-A$ generates the analytic semigroup $T$ and $H^{\infty }$ is the space of bounded analytic functions on a domain strictly containing the spectrum of $A$. We show that $b(\epsilon )=\mathcal{O}(|\log \epsilon |)$ 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 $b(\epsilon )=\mathcal{O}(\sqrt{|\log \epsilon |})$.

Preprint of the final, published version. In comparison with previous version, Prop. 2.2 was added and Thm. 3.5 has been slightly adapted in order to point out the major assertion

References in corpus (4)

• Functional calculus for semigroup generators via transference
• Convergence of subdiagonal Padé approximations of $C_0$-semigroups
• The Weiss conjecture and weak norms
• Functional calculus estimates for Tadmor-Ritt operators

Cited by in corpus (2)

• Functional calculus estimates for Tadmor-Ritt operators
• A Besov algebra calculus for generators of operator semigroups and related norm-estimates