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<∞, J p ,J p * respectively denote the duality mapping with gauge function φ(t)=t p−1 from X onto X * and S * onto X. If T:X→X is a bounded linear operator, then M(T):X→X is the mapping defined by M(T)=J q * T * J p T, where T * :S*→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 T n ≤1 for all n, then M(T n , ..., T 1 )x strongly converges in X for any x∈X, with an estimate of the rate of convergence : M(T n ...T 1 )x-M(x)≤σ(x)xΨ(1-(m(x)/T n ...T 1 x) p ), where M(x)=lim n→ ∞M(T n ...T 1 )x, m(x)=lim n→ ∞T n ...T 1 x, and σ:X→R + , Ψ:R + →R + are definite, strictly increasing positive functions. The result obtained generalizes and improves on the theorem recently by Akcoglu and Sucheston