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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

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


Authors: Zong Ben Xu and G. F. Roach
Journal: Proc. Amer. Math. Soc. 111 (1991), 1067-1074
MSC: Primary 47A99; Secondary 47B60
MathSciNet review: 1049854
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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\Vert {{T_n}} \right\Vert \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:

$\displaystyle \vert\vert M({T_n} \cdots {T_1})x - M(x)\vert\vert \leq \sigma (x... ...\vert x\vert\vert\psi (1 - {(m(x)/\vert\vert{T_n} \cdots {T_1}x\vert\vert)^p}),$

, where $ M(x) = {\lim _{n \to \infty }}M({T_n} \cdots {T_1})x,m(x) = {\lim _{n \to \infty }}\left\Vert {{T_n} \cdots {T_1}x} \right\Vert$, 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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1049854-3
PII: S 0002-9939(1991)1049854-3
Article copyright: © Copyright 1991 American Mathematical Society