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Archiv der Mathematik

, Volume 71, Issue 5, pp 388–398 | Cite as

Counterexamples concerning sectorial operators

  • Gilles Lancien
Article

Abstract.

In this paper we give two counterexamples to the closedness of the sum of two sectorial operators with commuting resolvents. In the first example the operators are defined on an L p-space, with \(1 \le p \neq 2 \le \infty \), and one of them admits bounded imaginary powers. The second example is concerned with operators defined on a Hilbert valued L p-space; one acts on L p and admits bounded imaginary powers as the other acts on the Hilbert space. In the last section of the paper we show that the two partial derivations on \(L^2 ({\Bbb R}^2;X)\) admit a so-called bounded joint functional calculus if and only if X is a UMD Banach space with property \((\alpha )\) (geometric property introduced by G. Pisier).

Keywords

Hilbert Space Banach Space Geometric Property Functional Calculus Sectorial Operator 

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Copyright information

© Birkhäuser Verlag, Basel 1998

Authors and Affiliations

  • Gilles Lancien
    • 1
  1. 1.Equipe de Mathématiques – UMR 6623, Université de Franche-Comté, F-25030 Besançon Cedex, FranceFrance

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