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

TL;DR

This paper explores the relationship between R-boundedness and gamma-boundedness in Banach spaces, showing that aside from known implications, no other inclusions hold, and characterizing when R-boundedness is stable under adjoints.

Contribution

It proves that all other potential inclusions between R-boundedness and gamma-boundedness fail, and characterizes K-convexity as the condition for R-boundedness stability under adjoints.

Findings

R-boundedness and gamma-boundedness coincide in Banach spaces with finite cotype.

R-boundedness implies gamma-boundedness, but not vice versa.

R-boundedness is stable under taking adjoints if and only if the space is K-convex.

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.