Alternating procedures in uniformly smooth Banach spaces

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 $({J_\mu }x,x) = \left \| {{J_\mu }x} \right \| \cdot \left \| x \right \|$ and $\left \| {{J_\mu }x} \right \| = \mu \left ( {\left \| x \right \|} \right )$. 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].