Orthogonal decompositions in Hilbert A-modules

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:470

Article

Cerreia-Vioglio, S.¹  Maccheroni, F.^1,2  Marinacci, M.^1,2 

[1] Univ Bocconi, Milan, Italy

[2] IGIER, Milan, Italy

关键词: Hilbert modules;    Orthogonal decompositions;    Martingale decompositions;    Stricker's lemma;   

DOI  :  10.1016/j.jmaa.2018.10.035

来源: Elsevier

【 摘 要 】

Pre-Hilbert A-modules are a natural generalization of inner product spaces in which the scalars are allowed to be from an arbitrary algebra. In this perspective, submodules are the generalization of vector subspaces. The notion of orthogonality generalizes in an obvious way too. In this paper, we focus on f-algebras A that are either of L-infinity or of L-0 type. We provide necessary and sufficient topological conditions for a submodule to be orthogonally complemented. Then, we present four applications of our results. The most important ones are Doob's and Kunita-Watanabe's decompositions for conditionally square-integrable processes. They are obtained as orthogonal decomposition results carried out in an opportune pre-Hilbert A-module. Second, we show that a version of Stricker's Lemma can be also derived as a corollary of our results. Finally, we provide a version of the Koopman von-Neumann decomposition theorem for a specific pre-Hilbert module which is useful in Ergodic Theory. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free