An alternating procedure for operators on uniformly convex and uniformly smooth Banach spaces

Xu, Zong Ben; Roach, G. F.

doi:10.1090/S0002-9939-1991-1049854-3

An alternating procedure for operators on uniformly convex and uniformly smooth Banach spaces
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by Zong Ben Xu and G. F. Roach PDF

Proc. Amer. Math. Soc. 111 (1991), 1067-1074 Request permission

Abstract:

Let $X$ be a real uniformly convex and uniformly smooth Banach space. For any $1 < p < \infty ,{J_p},J_p^ *$ respectively denote the duality mapping with gauge function $\varphi (t) = {t^{p - 1}}$ from $X$ onto ${X^ * }$ and ${X^*}$ onto $X$. If $T:X \to X$ is a bounded linear operator, then $M(T):X \to X$ is the mapping defined by $M(T) = J_q^ * {T^ * }{J_p}T$, where ${T^ * }:{X^ * } \to {X^ * }$ is the adjoint of $T$ and $q = {(p - 1)^{ - 1}}p$. It is proved that if ${T_n}$ is a sequence of operators on $X$ such that $\left \| {{T_n}} \right \| \leq 1$ for all $n$, then $M({T_n}, \ldots ,{T_1})x$ strongly converges in $X$ for any $x \in X$, with an estimate of the rate of convergence: \[ ||M({T_n} \cdots {T_1})x - M(x)|| \leq \sigma (x)||x||\psi (1 - {(m(x)/||{T_n} \cdots {T_1}x||)^p}),\], where $M(x) = {\lim _{n \to \infty }}M({T_n} \cdots {T_1})x,m(x) = {\lim _{n \to \infty }}\left \| {{T_n} \cdots {T_1}x} \right \|$, and $\sigma :X \to {R^ + },\psi :{R^ + } \to {R^ + }$ are definite, strictly increasing positive functions. The result obtained generalizes and improves on the theorem offered recently by Akcoglu and Sucheston [1].

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