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Optimal polynomial decay of functions and operator semigroups. (English) Zbl 1185.47044
Summary: We characterize the polynomial decay of orbits of Hilbert space $$C_{0}$$-semigroups in resolvent terms. We also show that results of the same type for general Banach space semigroups and functions obtained recently in C. J. K. Batty and T. Duyckaerts [J. Evol. Equ. 8, No. 4, 765–780 (2008; Zbl 1185.47043)] are sharp. This settles a conjecture posed by Batty and Duyckaerts in the aforementioned paper.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 34D05 Asymptotic properties of solutions to ordinary differential equations 46B20 Geometry and structure of normed linear spaces
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