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

Abstract:

For Banach spaces $X$ and $Y$ and a bounded linear operator $T:X \rightarrow Y$ we let $\rho (T):=\inf c$ such that \[ \left ( AV_{\theta _l = \pm 1} \left \|\sum \limits _{l=1}^\infty \theta _l \left ( \sum \limits _{k=\tau _{l-1}+1}^{\tau _l} h_k T x_k \right )\right \|_{L_2^Y}^2 \right )^{\frac {1}{2}} \le c \left \| \sum \limits _{k=1}^\infty h_k x_k \right \| _{L_2^X} \] for all finitely supported $(x_k)_{k=1}^\infty \subset X$ and all $0 = \tau _0 < \tau _1 < \cdots$, where $(h_k)_{k=1}^\infty \subset L_1[0,1)$ is the sequence of Haar functions. We construct an operator $T:X \rightarrow X$, where $X$ is superreflexive and of type 2, with $\rho (T)<\infty$ such that there is no constant $c>0$ with \[ \sup _{\theta _k = \pm 1} \left \| \sum \limits _{k=1}^\infty \theta _k h_k T x_k \right \| _{L_2^X} \le c \left \| \sum \limits _{k=1}^\infty h_k x_k \right \| _{L_2^X}. \]

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 $\operatorname {UMD}$–constants. As a by-product we extend the characterization of the non–superreflexive Banach spaces by the finite tree property using lower 2–estimates of sums of martingale differences.