Convergence of random products of compact contractions in Hilbert space

Abstract

Let {T₁, ..., T_N} be a finite set of linear contraction mappings of a Hilbert space H into itself, and let r be a mapping from the natural numbers N to {1, ..., N}. One can form S_n=T_r(n)...T_r(1) which could be described as a random product of the T_i's. Roughly, the S_n converge strongly in the mean, but additional side conditions are necessary to ensure uniform, strong or weak convergence. We examine contractions with three such conditions. (W): ‖x_n‖≤1, ‖Tx_n‖→1 implies (I-T)x_n→0 weakly, (S): ‖x_n‖≤1, ‖Tx_n‖→1 implies (I-T)x_n→0 strongly, and (K): there exists a constant K>0 such that for all x, ‖(I-T)x‖²≤K(‖x‖²−‖Tx‖²).

We have three main results in the event that the T_i's are compact contractions. First, if r assumes each value infinitely often, then S_n converges uniformly to the projection Q on the subspace ⋂_i= ^N₁ [x|T_ix=x]. Secondly we prove that for such compact contractions, the three conditions (W), (S), and (K) are equivalent. Finally if S=S(T₁, ..., T_N) denotes the algebraic semigroup generated by the T_i's, then there exists a fixed positive constant K such that each element in S satisfies (K) with that K.