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Proceedings of the American Mathematical Society

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



Alternating procedures in uniformly smooth Banach spaces

Author: I. Assani
Journal: Proc. Amer. Math. Soc. 104 (1988), 1131-1133
MSC: Primary 47A35; Secondary 46B20
MathSciNet review: 937842
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Abstract: Let $ E$ be a uniformly smooth Banach space and $ C$ the set of real continuous strictly increasing functions $ \mu $ on $ {{\mathbf{R}}_ + }$ such that $ \mu (0) = 0$. At each $ \mu $ we can associate a unique duality map $ {J_\mu }:E \to {E^ * }$ such that $ ({J_\mu }x,x) = \left\Vert {{J_\mu }x} \right\Vert \cdot \left\Vert x \right\Vert$ and $ \left\Vert {{J_\mu }x} \right\Vert = \mu \left( {\left\Vert x \right\Vert} \right)$. We prove in this note that if $ {T_n}$ is a sequence of linear contractions on $ E$ the sequence $ T_1^ * T_2^ * \cdots T_n^ * {J_\mu }{T_n} \cdots {T_2}{T_1}x$ converges strongly in $ {E^ * }$ norm for all $ x$ in $ E$. In particular if $ {E^ * }$ is also uniformly smooth then for any $ \mu $ and $ \nu $ in $ C$ the sequence $ J_\nu ^ * T_1^ * T_2^ * \cdots T_n^ * {J_\mu }{T_n} \cdots {T_1}x$ converges in $ E$ norm. This generalizes a result of M. Akcoglu and L. Sucheston [1].

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Keywords: Uniformly smooth Banach spaces, duality map, pointwise convergence
Article copyright: © Copyright 1988 American Mathematical Society