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

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    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 languageEnglish
    Pages (from-to)49-84
    Number of pages36
    JournalJournal of functional analysis
    Issue number1
    Publication statusPublished - Jul 2016


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