## Abstract

We study the functional calculus properties of generators of C _{0} -groups under type and cotype assumptions on the underlying Banach space. In particular, we show the following. Let -iA generate a C _{0} -group on a Banach space X with type p∈[1, 2] and cotype q ∈ [2,∞). Then f(A):(X,D(A)) _{1p-1q,1} → X is bounded for each bounded holomorphic function f on a sufficiently large strip. As a corollary of this result, for sectorial operators, we quantify the gap between bounded imaginary powers and a bounded ℋ ^{∞} -calculus in terms of the type and the cotype of the underlying Banach space. For cosine functions, we obtain similar results as for C _{0} -groups. We extend our theorems to R-bounded operator-valued calculi, and we give an application to the theory of rational approximation of C _{0} -groups.

Original language | English |
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Pages (from-to) | 17-47 |

Number of pages | 31 |

Journal | Quarterly Journal of Mathematics |

Volume | 70 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Mar 2019 |

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