El teorema de Amemiya y Ando para un producto aleatorio de operadores contractivos.

Abstract

Let H be a Hilbert space and T : H → H a contraction, i.e., a linear and bounded operator such that ||T|| ≤ 1. We say that T has the property (W) when for any sequence (fn)n∈N in H such that || fn|| ≤ 1 ∀ n ∈ N and límn→∞ ||T fn|| = 1, we have that límn→∞ (I − T)fn = 0 weakly. We consider contractions T1, T2, . . . , TN : H → H, which have the property (W), and a quasi-periodic function r : N → {1, 2, . . . , N} with quasi-period q, and sequences of operators (A(1)_n)n∈N, (A(2)_n)n∈N , . . . , (A(N)_n)n∈N from H to H such that ||A(k)_n x – T_kx|| ≤ γn||x|| ∀ x ∈ H, ∀ k = 1, . . . , N, where (γn)n∈N is a sequence of positive real numbers generating a convergent series. In the main result of this work we prove that for each x ∈ H there exists a point x ∗ ∈ H such that weakly.