Geometrical and Martingale characterizations of UMD and Hilbert spaces

Lee, Jinsik Mok

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https://hdl.handle.net/2142/23039 Copy

Description

Title

Geometrical and Martingale characterizations of UMD and Hilbert spaces

Author(s)

Lee, Jinsik Mok

Issue Date

1992

Doctoral Committee Chair(s)

Burkholder, Donald L.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

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).

Type of Resource

text

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http://hdl.handle.net/2142/23039

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Copyright 1992 Lee, Jinsik Mok

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