Unrestricted Products of Contractions in Banach Spaces

Let X be a reflexive Banach space such that for any x 6= 0 the set {x∗ ∈ X∗ : ‖x∗‖ = 1 and x∗(x) = ‖x‖} is compact. We prove that any unrestricted product of of a finite number of (W ) contractions on X converges weakly. Recall a (bounded) linear operator on a Banach space X is said to be contraction if ‖Tx‖ ≤ ‖x‖ for all x ∈ X . T is said to satisfy condition (W ) if {xn} is bounded and ‖xn‖ − ‖Txn‖ converges to 0 implies xn − Txn converges to 0 weakly. An algebraic semigroup S generated by a (possibly infinite) set of contractions is said to satisfy condition (W ) if for any bounded sequence of vectors {xn} ⊆ X and a sequence of words {Wn} from S such that ‖xn‖−‖Wnxn‖ converges to 0, xn−Wnxn converges to 0 weakly. Let {T1, T2, . . . , TN} be N (W )-contractions on X , and let r be a mapping from the set of natural numbers N onto {1, 2, . . . , N}, which assumes each value infinitely often. An unrestricted (or random) product of these operators is the sequence {Sn : n = 1, 2, . . . } defined by Sn = Tr(n)Tr(n−1) . . . Tr(1). John von Neumann [8] proved that if T1 and T2 are orthogonal projections on Hilbert space, and if {Sn} is a random product of {T1, T2}, then Sn converges strongly. Amemiya and Ando [1] extended this result by showing if {Sn} is a random product of finite (W )-contractions on Hilbert space, then Sn converges weakly. On the other hand, Bruck [2] showed that a random product of infinite (W )-contractions on Hilbert space is not necessary to be convergent weakly. One may ask the following question. Question 1. Does every random product of finite number of (W )-contractions on a reflexive Banach space X converge weakly? Recently, J. M. Dye, A. Khamsi, and S. Reich [4] showed the answer is positive if X is smooth. Indeed, they proved that if X is a smooth reflexive Banach space, and {T1, T2, · · · , TN} are N (W )-contractions on X , then (i) there is a contraction projection Q of X onto the common fixed point set such that QTj = TjQ for 1 ≤ j ≤ N ; 1991 Mathematics Subject Classification. 47A05, 47B05, 65J10.