$R$-boundedness versus $γ$-boundedness

It is well-known that in Banach spaces with finite cotype, the $R$-bounded and $γ$-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 $γ$-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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