26 research outputs found

    Generation of subordinated holomorphic semigroups via Yosida's theorem

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    Using functional calculi theory, we obtain several estimates for ψ(A)g(A)\|\psi(A)g(A)\|, where ψ\psi is a Bernstein function, gg is a bounded completely monotone function and A-A is the generator of a holomorphic C0C_0-semigroup on a Banach space, bounded on [0,)[0,\infty). Such estimates are of value, in particular, in approximation theory of operator semigroups. As a corollary, we obtain a new proof of the fact that ψ(A)-\psi(A) generates a holomorphic semigroup whenever A-A does, established recently in [8] by a different approach

    Product formulas in functional calculi for sectorial operators

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    We study the product formula (fg)(A)=f(A)g(A)(fg)(A) = f(A)g(A) in the framework of (unbounded) functional calculus of sectorial operators AA. We give an abstract result, and, as corollaries, we obtain new product formulas for the holomorphic functional calculus, an extended Stieltjes functional calculus and an extended Hille-Phillips functional calculus. Our results generalise previous work of Hirsch, Martinez and Sanz, and Schilling.Comment: This is the authors accepted manuscript for a paper being published in Mathematische Zeitschrift. The final publication is available at Springer via http://dx.doi.org/10.1007/s00209-014-1378-

    Sarnak's conjecture implies the Chowla conjecture along a subsequence

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    We show that the M\"obius disjointess of zero entropy dynamical systems implies the existence of an increasing sequence of positive integers along which the Chowla conjecture on autocorrelations of the M\"obius function holds.Comment: Proceedings of the Chair Morlet semester "Ergodic Theory and Dynamical Systems in their Interactions with Arithmetic and Combinatorics" 1.08.2016--31.01.2017, Springer, to appea

    Resolvent representations for functions of sectorial operators

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    We obtain integral representations for the resolvent of ψ(A)\psi(A), where ψ\psi is a holomorphic function mapping the right half-plane and the right half-axis into themselves, and AA is a sectorial operator on a Banach space. As a corollary, for a wide class of functions ψ\psi, we show that the operator ψ(A)-\psi(A) generates a sectorially bounded holomorphic C0C_0-semigroup on a Banach space whenever A-A does, and the sectorial angle of AA is preserved. When ψ\psi is a Bernstein function, this was recently proved by Gomilko and Tomilov, but the proof here is more direct. Moreover, we prove that such a permanence property for AA can be described, at least on Hilbert spaces, in terms of the existence of a bounded HH^{\infty}-calculus for AA. As byproducts of our approach, we also obtain new results on functions mapping generators of bounded semigroups into generators of holomorphic semigroups and on subordination for Ritt operators.Comment: The paper has been accepted for publication in Advances in Mathematics. This is the authors' accepted versio

    A Besov algebra calculus for generators of operator semigroups and related norm-estimates

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    We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille-Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.Comment: This is the authors' accepted version of a paper which will be published in Mathematische Annale