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

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:

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

, 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].