Abstract
We study sectorial operators with a special type of functional calculus, which we term an absolute functional calculus. A typical example of such an operator is an invertible operator A (defined on a Banach space X) considered on the real interpolation space (Dom(A), X)θ,p where 0 < θ < 1 and 1 < p < ∞. In general the absolute functional calculus can be characterized in terms of real interpolation spaces. We show that operators of this type have a strong form of the H ∞-calculus and behave very well with respect to the joint functional calculus. We give applications of these results to recent work of Arendt, Batty and Bu on the existence of Hölder-continuous solutions for the abstract Cauchy problem.
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The authors acknowledge support from NSF grant DMS-0244515.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Kalton, N.J., Kucherenko, T. Operators with an absolute functional calculus. Math. Ann. 346, 259–306 (2010). https://doi.org/10.1007/s00208-009-0399-4
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DOI: https://doi.org/10.1007/s00208-009-0399-4