\(R\)-boundedness versus \(\gamma\)-boundedness (Q283981)

The authors consider the connection between the notions of \(R\)-boundedness, \(\gamma\)-boundedness and \(\ell^2\)-boundedness (in the case of Banach lattices) of families of operators acting between Banach spaces. Recall that a family \(\mathcal F\subset L(X,Y)\) of bounded linear operators acting between two Banach spaces \(X\) and \(Y\) is called \(R\)-bounded if there exists a constant \(C>0\) such that NEWLINE\[NEWLINE\Big(\mathbb E \|\sum_{n01}^N r_n T_nx_n\|^2\Big)^{1/2}\leq C \Big(\mathbb E \|\sum_{n=1}^N r_n x_n\|^2\Big)^{1/2}NEWLINE\]NEWLINE for all \(N\geq 1\), all \((x_n)_{n=1}^N\subset X\) and \((T_n)_{n=1}^N \subset \mathcal F\), where \((r_n)\) stands for the Rademacher sequence. The notion of \(\gamma\)-bounded is analogous, replacing the Rademacher sequence by a sequence of independent standard Gaussian random variables \(\gamma_n\). In the case of Banach lattices \(X\) and \(Y\), the family is called \(\ell^2\)-bounded whenever there exists a constant \(C>0\) such that NEWLINE\[NEWLINE\|(\sum_{n=1}^N|T_nx_n|^2)^{1/2}\|\leq C \|(\sum_{n=1}^N|x_n|^2)^{1/2}\|NEWLINE\]NEWLINE for all \(N\geq 1\), all \((x_n)_{n=1}^N\subset X\) and \((T_n)_{n=1}^N \subset \mathcal F\). It is well known that \(R\)-boundedness implies \(\gamma\)-boundedness and that both notions coincide when \(X\) is assumed to have finite cotype.NEWLINENEWLINEThe main result of the paper establishes that this is actually the only situation where this happens, showing that if any \(\gamma\)-bounded family is also \(R\)-bounded, then \(X\) must have finite cotype. In the case of Banach lattices, the authors show that \(\ell^2\)-boundedness implies \(R\)-boundedness if and only if \(Y\) has finite cotype and that \(R\)-boundedness implies \(\ell^2\)-boundedness if and only if \(X\) has finite cotype. Although their results, as pointed out by the authors, could be achieved from some deep results by Montgomery-Smith and Talagrand on cotype of operators from \(C(K)\), they present elementary and independent proofs. Also, they characterize when \(R\)-boundedness and \(\gamma\)-boundedness are stable under taking adjoints.