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Transactions of the American Mathematical Society

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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-constants
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by Stefan Geiss PDF
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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Additional Information
  • Stefan Geiss
  • Affiliation: Mathematisches Institut der Friedrich–Schiller–Universität, Postfach, D–O7740 Jena, Germany
  • MR Author ID: 248903
  • Email: geiss@minet.uni-jena.de
  • Received by editor(s): November 4, 1996
  • Received by editor(s) in revised form: April 8, 1997
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 1355-1375
  • MSC (1991): Primary 46B07, 60G42; Secondary 46B70, 60B11
  • DOI: https://doi.org/10.1090/S0002-9947-99-02093-0
  • MathSciNet review: 1458301