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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Characterization of conditional expectation operators for Banach-valued functions


Authors: D. Landers and L. Rogge
Journal: Proc. Amer. Math. Soc. 81 (1981), 107-110
MSC: Primary 60B11; Secondary 60A05
DOI: https://doi.org/10.1090/S0002-9939-1981-0589148-7
MathSciNet review: 589148
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Abstract: Let $B$ be a Banach space. It is well known that for every probability space $(\Omega , \mathcal {A}, P)$ and every sub-$\sigma$-field $\mathcal {A}_0 \subset \mathcal {A}$ there exists a conditional expectation operator $P^{\mathcal {A}_0}X$ for $B$-valued $P$-integrable functions. This operator maps the space ${L_p}(\Omega , \mathcal {A}, P, B)$ into itself for each $p \geqslant 1$. The operator is linear, idempotent, constant preserving and contractive in ${L_p}$. For $B = {\mathbf {R}}$ and $p \ne 2$ these conditions characterize a conditional expectation operator. It turns out that in general these properties characterize conditional expectation operators for Banach-valued functions only for $p = 1$ and strictly convex Banach spaces.


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Keywords: Conditional expectation, contractive operators, uniformly convex Banach spaces, strictly convex Banach spaces, <IMG WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${L_p}$">-spaces
Article copyright: © Copyright 1981 American Mathematical Society