Alternating procedures in uniformly smooth Banach spaces

Abstract

Let E E be a uniformly smooth Banach space and C C the set of real continuous strictly increasing functions μ \mu on R + {{\mathbf {R}}_ + } such that μ ( 0 ) = 0 \mu (0) = 0 . At each μ \mu we can associate a unique duality map J μ : E → E ∗ {J_\mu }:E \to {E^ * } such that ( J μ x , x ) = ‖ J μ x ‖ ⋅ ‖ x ‖ ({J_\mu }x,x) = \left \| {{J_\mu }x} \right \| \cdot \left \| x \right \| and ‖ J μ x ‖ = μ ( ‖ x ‖ ) \left \| {{J_\mu }x} \right \| = \mu \left ( {\left \| x \right \|} \right ) . We prove in this note that if T n {T_n} is a sequence of linear contractions on E E the sequence T 1 ∗ T 2 ∗ ⋯ T n ∗ J μ T n ⋯ T 2 T 1 x T_1^ * T_2^ * \cdots T_n^ * {J_\mu }{T_n} \cdots {T_2}{T_1}x converges strongly in E ∗ {E^ * } norm for all x x in E E . In particular if E ∗ {E^ * } is also uniformly smooth then for any μ \mu and ν \nu in C C the sequence J ν ∗ T 1 ∗ T 2 ∗ ⋯ T n ∗ J μ T n ⋯ T 1 x J_\nu ^ * T_1^ * T_2^ * \cdots T_n^ * {J_\mu }{T_n} \cdots {T_1}x converges in E E norm. This generalizes a result of M. Akcoglu and L. Sucheston [1].