A Besov algebra calculus for generators of operator semigroups and related norm-estimates
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Mathematische Annalen
A Besov algebra calculus for generators of operator semigroups and related norm-estimates Charles Batty1 · Alexander Gomilko2 · Yuri Tomilov3 Received: 28 January 2019 / Revised: 7 October 2019 © The Author(s) 2019
Abstract 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. Mathematics Subject Classification Primary 47A60; Secondary 30H25 · 46E15 · 47D03
Communicated by Y. Giga. This work was partially supported financially by a Leverhulme Trust Visiting Research Professorship and an NCN grant UMO-2017/27/B/ST1/00078, and inspirationally by the ambience of the Lamb and Flag, Oxford. We are very grateful to two referees for helpful comments on the first version of this paper, in particular those relating to Sects. 4.4 and 4.5.
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Yuri Tomilov [email protected] Charles Batty [email protected] Alexander Gomilko [email protected]
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St. John’s College, University of Oxford, Oxford OX1 3JP, UK
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Faculty of Mathematics and Computer Science, Nicolas Copernicus University, Chopin Street 12/18, 87-100 Toru´n, Poland
3
´ Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-956 Warsaw, Poland
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C. Batty et al.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Besov algebra, related spaces and a duality . . . . . . . 3 Norm-estimates for some subclasses of B . . . . . . . . . . 4 Functional calculus for B . . . . . . . . . . . . . . . . . . 5 Applications of the B-calculus for operator norm-estimates 6 Appendix: Relation to Besov spaces . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction Given a linear operator A on a Banach space X , a fundamental matter in operator theory is to define a functional calculus for A and to get reasonable norm-estimates for functions of A. A rich enough functional calculus for A yields various spectral decomposition properties and leads to a detailed spectral theory. One well-known instance of that is the functional calculus for normal operators on Hilbert spaces. A common procedure to create a functional calculus for an unbounded operator A is to define a function algebra A on the spectrum σ (A) of A and a homomorphism from A into the space of bounded
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