### Abstract

We investigate the boundedness of the H$^\infty$-calculus by estimating the bound b(ε) of the mapping H$^\infty$→B(X): f→f(A)T(ε) for ε 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(ε) =O(|logε|) 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(|logε|).

Original language | English |
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Pages (from-to) | 49-84 |

Number of pages | 36 |

Journal | Journal of functional analysis |

Volume | 271 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 2016 |