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

Abstract:

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