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NEW COUNTEREXAMPLES ON RITT OPERATORS, SECTORIAL OPERATORS AND $R$-BOUNDEDNESS

Part of: Normed linear spaces and Banach spaces; Banach lattices General theory of linear operators Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  11 April 2019

LORIS ARNOLD  and
CHRISTIAN LE MERDY Open the ORCID record for CHRISTIAN LE MERDY [Opens in a new window]
LORIS ARNOLD
Affiliation:
Laboratoire de Mathématiques de Besançon, UMR 6623, CNRS, Université Bourgogne Franche-Comté, 25030 Besançon Cedex, France email loris.arnold@univ-fcomte.fr
CHRISTIAN LE MERDY*
Affiliation:
Laboratoire de Mathématiques de Besançon, UMR 6623, CNRS, Université Bourgogne Franche-Comté, 25030 Besançon Cedex, France email clemerdy@univ-fcomte.fr
*
✉clemerdy@univ-fcomte.fr
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Abstract

Let ${\mathcal{D}}$ be a Schauder decomposition on some Banach space $X$. We prove that if ${\mathcal{D}}$ is not $R$-Schauder, then there exists a Ritt operator $T\in B(X)$ which is a multiplier with respect to ${\mathcal{D}}$ such that the set $\{T^{n}:n\geq 0\}$ is not $R$-bounded. Likewise, we prove that there exists a bounded sectorial operator $A$ of type $0$ on $X$ which is a multiplier with respect to ${\mathcal{D}}$ such that the set $\{e^{-tA}:t\geq 0\}$ is not $R$-bounded.

Keywords

sectorial operatorsRitt operatorsR-boundedness

MSC classification

Primary: 47D03: Groups and semigroups of linear operators
Secondary: 46B15: Summability and bases 47A99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 3 , December 2019 , pp. 498 - 506
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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Footnotes

The authors were supported by the French ‘Investissements d’Avenir’ program, project ISITE-BFC (contract ANR-15-IDEX-03).

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