Files in this item



application/pdf9215847.pdf (2MB)Restricted to U of Illinois
(no description provided)PDF


Title:Geometrical and Martingale characterizations of UMD and Hilbert spaces
Author(s):Lee, Jinsik Mok
Doctoral Committee Chair(s):Burkholder, Donald L.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:Suppose that X is a real or complex Banach space with norm $\vert \cdot \vert$. Then X is a Hilbert space if and only if $E\vert x + Y\vert \geq 1$ for all x $\in$ X and all X-valued Bochner integrable functions Y on the Lebesgue unit interval satisfying EY = 0 and $\vert Y\vert \geq$ 1 a.e. This leads to a simple proof of the biconvex-function characterization due to Burkholder.
There is a dual result. The Banach space X is a Hilbert space if and only if $E\vert x + Y\vert \leq$ 2 for all x in X with $\vert x\vert$ $\leq$ 2 and all Y satisfying EY = 0 and $\vert x - Y\vert$ $\leq$ 2 a.e. This leads to the following biconcave-function characterization: A Banach space X is a Hilbert space if and only if there is a biconcave function $\eta:\{(x,y) \in {\bf X} \times {\bf X}:\vert x - y\vert \leq 2\} \to$ R such that $\eta$(0,0) = 2 and $\eta(x,y) \geq \vert x + y\vert$. If the condition $\eta$(0,0) = 2 is replaced by the condition $\eta$(0,0) $<$ $\infty$, then the existence of such a function $\eta$ characterizes UMD (Banach spaces with the unconditionality property for martingale differences).
Issue Date:1992
Rights Information:Copyright 1992 Lee, Jinsik Mok
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9215847
OCLC Identifier:(UMI)AAI9215847

This item appears in the following Collection(s)

Item Statistics