A generalization of a theorem of Amemiya and Ando on the convergence of random products of 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 numbersN to {1, ..., N} which assumes each value infinitely often. One can form S_n=T_r(n)...T_r(1) which could be described as a random product of the T_i's. If the contractions have the condition (W): ‖Tx‖<‖x‖ whenever Tx≠x, then S_n converges weakly to the projection Q onto the subspace$\cap _{i = \mathop 1\limits^N } [x|T_i x = x]$. This theorem is due to Amemiya and Ando. We demonstrate a basic property of the algebraic semigroupS=S(T₁, ..., T_N) generated by N contractions, each having (W). We prove that if the semigroup of an infinite set of contractions is equipped with this property, and the maps satisfy a minor condition parallel to (W) on each of N maps, then random products still converge weakly. Our proof is different from Amemiya and Ando's. We illustrate our method with a new proof of the fact that if a contraction T is completely non-unitary, then Tⁿ→0 weakly.