A counterexample concerning the relation between decoupling constants and UMD-constants

For Banach spaces X and Y and a bounded linear operator T : X --> Y we let rho(T) := inf c such that [GRAPHICS] for all finitely supported (x(k))(k=1)(infinity) subset of X and all 0 = ta u(0) < tau(1) < ..., where (h(k))(k-1)(infinity) subset of L-1[0; 1) is the sequence of Haar functions. We construct an operator T : X --> X, where X is superreflexive and of type 2, with rho(T) < 1 such that there is no cons tant c > 0 with [GRAPHICS] In particular it turns out that the decoupling constants rho(I-X), where I- X is the identity of a Banach space X, fail to be equivalent up to absolute multiplicative constants to the usual UMD-constants. As a by-product we ex tend the characterization of the non-superreflexive Banach spaces by the fi nite tree property using lower 2-estimates of sums of martingale difference s.