Type and Cotype with Respect to Arbitrary Orthonormal Systems Original Research Article

Let Φ=(φk)k ∈ N be an orthonormal system on some σ-finite measure space (Ω,p). We study the notion of cotype with respect to Φ for an operator T between two Banach spaces X and Y, defined by cΦ(T)≔inf c such that [formula] where (gk)(k ∈ N) is the sequence of independent and normalized gaussian variables. It is shown that this Φ-cotype coincides with the usual notion of cotype 2 iff c(Φ)(Iln∞ ∼ [formula] uniformly in n iff there is a positive η > 0 such that for all n ∈ N one can find an orthonormal Ψ=(ψl)n1 ⊂ span {φk|k ∈ N} and a sequence of disjoint measurable sets (Al)n1 ⊂ Ω with ∫Al|ψl|2dp ≥ η for all l = 1, ..., n. A similar result holds for the type situation. The study of type and cotype with respect to orthonormal systems of a given length provides the appropriate approach to this result. We intend to give a quite complete picture for orthonormal systems in measure space with few atoms.