Rademacher and Gaussian averages and Rademacher cotype of operators between Banach spaces

A basic result of B. Maurey and G. Pisier states that Gaussian and Rademacher averages in a Banach space X X are equivalent if and only if X X has finite cotype. We complement this for linear bounded operators between Banach spaces. For T ∈ L ( X , Y ) T\in {{\mathcal L}}(X,Y) , let ϱ ( T | G n , R n ) \varrho (T|{\mathcal G}_n,{\mathcal R}_n) be the least c c such that \[ ( E ‖ ∑ k = 1 n T x k g k ‖ 2 ) 1 / 2 ≤ c ( E ‖ ∑ k = 1 n x k r k ‖ 2 ) 1 / 2 , \left ( {\mathbf E} \| \sum _{k=1}^n Tx_k g_k\|^2 \right )^{1/2} \le c \left ( {\mathbf E} \| \sum _{k=1}^n x_k r_k\|^2 \right )^{1/2}, \] where G n = ( g 1 , … , g n ) {\mathcal G}_n=(g_1,\ldots ,g_n) and R n = ( r 1 , … , r n ) {\mathcal R}_n=(r_1,\ldots ,r_n) are systems of n n independent standard Gaussian and Rademacher variables, respectively. Let ϱ ( T | I n , R n ) \varrho (T|{\mathcal I}_n,{\mathcal R}_n) be the Rademacher cotype 2 norm of T T computed with n n vectors. We prove inequalities showing that the asymptotic behaviour of the sequence ϱ ( T | G n , R n ) \varrho (T|{\mathcal G}_n,{\mathcal R}_n) is almost determined by the asymptotic behaviour of the sequence ϱ ( T | I n , R n ) \varrho (T|{\mathcal I}_n,{\mathcal R}_n) . In particular, we get \[ ϱ ( T | G n , R n ) = o ( 1 + log ⁡ n ) if and only if ϱ ( T | I n , R n ) = o ( n ) . \varrho (T|{\mathcal G}_n,{\mathcal R}_n) = o(\sqrt {1+\log n}) \mbox {if and only if} \varrho (T|{\mathcal I}_n,{\mathcal R}_n) = o(\sqrt {n}). \]