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A counterexample concerning the relation between decoupling constants and $\operatorname{UMD}$-constants

Author: Stefan Geiss
Journal: Trans. Amer. Math. Soc. 351 (1999), 1355-1375
MSC (1991): Primary 46B07, 60G42; Secondary 46B70, 60B11
MathSciNet review: 1458301
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Abstract: For Banach spaces $X$ and $Y$ and a bounded linear operator
$T:X \rightarrow Y$ we let $\rho(T):=\inf c$ such that

\begin{displaymath}\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} \end{displaymath}

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

\begin{displaymath}\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}. \end{displaymath}

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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Additional Information

Stefan Geiss
Affiliation: Mathematisches Institut der Friedrich–Schiller–Universität, Postfach, D–O7740 Jena, Germany

Keywords: Vector valued martingales, unconditional constants, superreflexive Banach spaces, interpolation of Banach spaces
Received by editor(s): November 4, 1996
Received by editor(s) in revised form: April 8, 1997
Article copyright: © Copyright 1999 American Mathematical Society