Comparing gaussian and Rademacher cotype for operators on the space of continous functions

Abstract

We will prove an abstract comparision principle which translates gaussian cotype in Rademacher cotype conditions and vice versa. More precisely, let $2\!<\!q\!<\!\infty$ and $T:\,C(K)\,\to\,F$ a linear, continous operator. T is of gaussian cotype q if and only if ( \summ_1^n (\frac{|| Tx_k||_F}{\sqrt{\log(k+1)}})^q )^{1/q} \, \le c || \summ_1^n \varepsilon_k x_k ||_{L_2(C(K))} , for all sequences with $(|| Tx_k ||)_1^n$ decreasing. T is of Rademacher cotype q if and only if (\summ_1^n (|| Tx_k||_F \,\sqrt{\log(k+1)})^q )^{1/q} \, \le c || \summ_1^n g_k x_k ||_{L_2(C(K))} , for all sequences with $(||Tx_k ||)_1^n$ decreasing. Our methods allows a restriction to a fixed number of vectors and complements the corresponding results of Talagrand.