# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## A counterexample concerning the relation between decoupling constants and UMD-constantsHTML articles powered by AMS MathViewer

by Stefan Geiss
Trans. Amer. Math. Soc. 351 (1999), 1355-1375 Request permission

## 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.

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