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Proceedings of the American Mathematical Society
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
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0589148-7
PII: S 0002-9939(1981)0589148-7
Keywords: Conditional expectation, contractive operators, uniformly convex Banach spaces, strictly convex Banach spaces, $ {L_p}$-spaces
Article copyright: © Copyright 1981 American Mathematical Society