Files in this item
Files  Description  Format 

application/pdf 9215847.pdf (2MB)  (no description provided) 
Description
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 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
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 Xvalued 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 biconvexfunction 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 biconcavefunction 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 
Type:  Text 
Language:  English 
URI:  http://hdl.handle.net/2142/23039 
Rights Information:  Copyright 1992 Lee, Jinsik Mok 
Date Available in IDEALS:  20110507 
Identifier in Online Catalog:  AAI9215847 
OCLC Identifier:  (UMI)AAI9215847 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois