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New conterexample on Ritt operators and \mathcal{R}-boundedness

publié le

Loris Arnold
(Université de Franche-Comté)

Let T be a bounded operator on a Banach space X. We say that T is a (\mathcal{R}-)Ritt operator if it satisfies that the two sequence (T^n)_{n \geq 0} and (nT^n(T - I))_{n\geq 0} are (\mathcal{R}-)bounded. It is possible to construct a Ritt operator which is not \mathcal{R}-Ritt, but until now, without knowing whether (T^n)_{n \geq 0} or (nT^n(T - I))_{n\geq 0} (or both) is not \mathcal{R}-bounded. We will give some preliminaries about the topic and we will construct a Ritt operator such that the sequence (T^n)_{n \geq 0} is not \mathcal{R}-bounded.