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 < infinity, J(p), J(p)* respectively denote the duality mapping with gauge function phi(t) = t(p)-1 from X onto X* and X* 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* : X* --> X* is the adjoint of T and q = (p-1)-1p. It is proved that if T(n) is a sequence of operators on X such that parallel-to-T(n)-parallel-to less-than-or-equal-to 1 for all n, then M(T(n),..., T1)chi strongly converges in X for any chi epsilon X, with an estimate of the rate of convergence: parallel-to-M(T(n)...T1)x - M(x)-parallel-to less-than-or-equal-to sigma (x) parallel-to-x-parallel-to psi (1 - (m(x)/parallel-to-T(n)...T1-x-parallel-to)p), where M(x) = lim(n) --> infinity M(T(n)...T1)x, m(x) = lim(n) --> infinity parallel-to T(n)...T1x-parallel-to, and sigma: X --> R+, psi: R+ --> R+ are definite, strictly increasing positive functions. The result obtained generalizes and improves on the theorem offered recently by Akcoglu and Sucheston [1].