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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

An alternating procedure for operators on $ L\sb p$ spaces


Authors: M. A. Akcoglu and L. Sucheston
Journal: Proc. Amer. Math. Soc. 99 (1987), 555-558
MSC: Primary 47A35; Secondary 28D99, 47B38
MathSciNet review: 875396
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Abstract: Let $ {L_p}$ be the usual Banach spaces over a $ \sigma $-finite measure space. If $ 1{\text{ < }}p{\text{ < }}\infty $ and $ q = p{(p - 1)^{ - 1}}$, then $ {\psi _p}:{L_p} \to {L_q}$ denotes the duality mapping defined by the requirements that $ (f,{\psi _p}f) = \left\Vert f \right\Vert _p^p = {\left\Vert f \right\Vert _p}\left\Vert {{\psi _p}f} \right\Vert q,f \in {L_p}$. If $ T:{L_p} \to {L_p}$ is a bounded linear operator, then $ M(T):{L_p} \to {L_p}$ is the mapping defined by $ M(T) = {\psi _q}{T^ * }{\psi _p}T$, where $ {T^ * }:{L_q} \to {L_q}$ is the adjoint of $ T$. It is proved that if $ {T_n}$ is a sequence of operators on $ {L_p}$ such that $ \left\Vert {{T_n}} \right\Vert \leq 1$ for all $ n$, then $ M({T_n} \cdots {T_2}{T_1})f$ converges in $ {L_p}$ for all $ f \in {L_p}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0875396-0
PII: S 0002-9939(1987)0875396-0
Keywords: Duality map on $ {L_p}$, contractions on $ {L_p}$
Article copyright: © Copyright 1987 American Mathematical Society