TYPE AND COTYPE WITH RESPECT TO ARBITRARY ORTHONORMAL SYSTEMS

Let phi=(phi(k))(k is an element of N) be an orthonormal system on som e sigma-finite measure space (Omega,p). We study the notion of cotype with respect to phi for an operator T between two Banach spaces X and Y, defined by c(phi)(T):=inf c such that [GRAPHICS] where (g)(k is an element of N) is the sequence of independent and normalized gaussian v ariables. It is shown that this phi-cotype coincides with the usual no tion of cotype 2 iff c(phi)(I(l infinity n)similar to root n/(log(n+1) ) uniformly in n iff there is a positive eta > 0 such that for all n i s an element of N one can find an orthonormal Psi=(psi(l))(1)(n) subse t of span {phi(k)\k is an element of N} and a sequence of disjoint mea surable sets (Al)(l)(n) subset of Omega with integral(Al)\psi l\(2) dp greater than or equal to eta for all I = 1, ..., n. A similar result holds for the type situation. The study of type and cotype with respec t to orthonormal systems of a given length provides the appropriate ap proach to this result. We intend to give a quite complete picture for orthonormal systems in measure space with few atoms. (C) 1995 Academic Press, Inc.