$R$ -Boundedness versus $\gamma$ -boundedness

Stanislaw Kwapień Institute of Mathematics, Warsaw University Mark Veraar Delft Institute of Applied Mathematics, Delft University of Technology Lutz Weis Institut für Analysis, Karlsruhe Institute of Technology

Analysis of PDEs Functional Analysis mathscidoc:1701.03034

Arkiv for Matematik, 54, (1), 125-145, 2014.11
It is well-known that in Banach spaces with finite cotype, the $R$ -bounded and $\gamma $ -bounded families of operators coincide. If in addition $X$ is a Banach lattice, then these notions can be expressed as square function estimates. It is also clear that $R$ -boundedness implies $\gamma $ -boundedness. In this note we show that all other possible inclusions fail. Furthermore, we will prove that $R$ -boundedness is stable under taking adjoints if and only if the underlying space is $K$ -convex.
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  title={ $R$ -Boundedness versus $\gamma$ -boundedness},
  author={Stanislaw Kwapień, Mark Veraar, and Lutz Weis},
  booktitle={Arkiv for Matematik},
Stanislaw Kwapień, Mark Veraar, and Lutz Weis. $R$ -Boundedness versus $\gamma$ -boundedness. 2014. Vol. 54. In Arkiv for Matematik. pp.125-145. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203642239366096.
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