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 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 Urbana-Champaign 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 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 Type: Text Language: English URI: http://hdl.handle.net/2142/23039 Rights Information: Copyright 1992 Lee, Jinsik Mok Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI9215847 OCLC Identifier: (UMI)AAI9215847
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