Abstract
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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The first named author is supported by NCN grant Dec-2012/05/B/ST1/00412. The second author is supported by the VIDI subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO). The third named author is supported by a grant from the Graduierten Kolleg 1294DFG.
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Kwapień, S., Veraar, M. & Weis, L. \(R\)-Boundedness versus \(\gamma\)-boundedness. Ark Mat 54, 125–145 (2016). https://doi.org/10.1007/s11512-015-0223-1
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DOI: https://doi.org/10.1007/s11512-015-0223-1