A generalization of a theorem of Amemiya and Ando on the convergence of random products of contractions in Hilbert space (Q582554)

Let \(\{T_ 1,T_ 2,...\}\) be a set of contractions on a Hilbert space. Let r be a mapping from the set of natural numbers into itself which assumes each range value infinitely often, and let \(S_ n=T_{r(n)}T_{r(n-1)}...T_{r(1)}\) for \(n=1,2,... \). The main result in the paper gives a sufficient condition for weak convergence of the sequence \(\{S_ n\}\). A discussion in the paper shows that the result generalizes a theorem of \textit{I. Amemiya} and \textit{T. Ando} [Acta Sci. Math. (Szeged) 26, 239-244 (1965; Zbl 0143.162)].