On measuring unboundedness of the H∞-calculus for generators of analytic semigroups

We investigate the boundedness of the H∞-calculus by estimating the bound b(ε) of the mapping H∞→B(X): f↦f(A)T(ε) for ε near zero. Here, −A generates the analytic semigroup T and H∞ is the space of bounded analytic functions on a domain strictly containing the spectrum of A. We show that b(ε)=O(|log⁡ε|) in general, whereas b(ε)=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(ε)=O(|log⁡ε|).