COMPARING GAUSSIAN AND RADEMACHER COTYPE FOR OPERATORS ON THE SPACE OF CONTINUOUS-FUNCTIONS

We prove an abstract comparison principle which translates gaussian co type into Rademacher cotype conditions and vice versa. More precisely, let 2 < q < infinity and T: C(K) --> F a continuous linear operator. (1) T is of gaussian cotype q if and only if ((k) Sigma (\\Tx(k)\\(F)\ root log(k+1))(q))(1/q) less than or equal to c\\(k) Sigma is an eleme nt of(k)x(k)\\(L2(C(K))), for all sequences (x(k))(k is an element of N) subset of C(K) with (\\Tx(k)\\)(k=1)(n) decreasing. (2) T is of Rad emacher cotype q if and only if ((k) Sigma(\\Tx(k)\\(F) root log(k+1)) (q))(1/q) less than or equal to c\\(k) Sigma gk(x)k\\(L2(C(K))), for a ll sequences (x(k))(k is an element of N) subset of C(K) with (\\Tx(k) \\)(k=1)(n) decreasing. Our method allows a restriction to a fixed num ber of vectors and complements the corresponding results of Talagrand.