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

Abstract

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.

The ℓ s-Boundedness of a Family of Integral Operators on UMD Banach Function Spaces

“…For Banach function spaces the notion was introduced in [40] under the name R s -boundedness, underlining its connection to the more well-known notion of R-boundedness. An extensive study of ℓ s -boundedness can be found in [24] and for a comparison between ℓ 2 -boundedness and R-boundedness we refer to [25].…”

Stability theory for semigroups using (L,L) Fourier multipliers

“…By replacing the Rademacher random variables in (2.1) by Gaussian variables, one obtains the definition of a γ-bounded collection T ⊆ L(X, Y ). Each R-bounded collection is γ-bounded, and the converse holds if and only if X has finite cotype (see [35,Theorem 1.1]). We choose to work with R-boundedness in this article, both because the notion of R-boundedness is more established and because those stability theorems in this article which use R-boundedness are only of interest on spaces with finite cotype.…”

Fourier multipliers in Banach function spaces with UMD concavifications

“…For a detailed treatment of R-boundedness we refer the reader to [23,29], and for ℓ s -boundedness see [28,50]. For R-and ℓ 2 -boundedness it suffices to consider subsets of T in the defining inequality (see [12,31]). For ℓ s -and ℓ r (ℓ s )-boundedness with r, s = 2 this is not the case: one must consider sequences, allowing for repeated elements.…”