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# Alternating Procedures in Uniformly Smooth Banach Spaces

I. Assani
Proceedings of the American Mathematical Society
Vol. 104, No. 4 (Dec., 1988), pp. 1131-1133
DOI: 10.2307/2047603
Stable URL: http://www.jstor.org/stable/2047603
Page Count: 3
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 $J_\mu: E \rightarrow E^\ast$ such that $(J_\mux, x) = \|J_\mu x\| \cdot \|x\|$ and $\|J_\mu x\| = \mu(\|x\|)$. We prove in this note that if $T_n$ is a sequence of linear contractions on $E$ the sequence $T^\ast_1T^\ast_2 \cdots T^\ast_nJ_\mu T_n \cdots T_2T_1x$ converges strongly in $E^\ast$ norm for all $x$ in $E$. In particular if $E^\ast$ is also uniformly smooth then for any $\mu$ and $\nu$ in $C$ the sequence $J^\ast_\nu T^\ast_1T^\ast_2 \cdots T^\ast_nJ_\mu T_n \cdots T_1x$ converges in $E$ norm. This generalizes a result of M. Akcoglu and L. Sucheston [1].