Alternating sequences with nonpositive operators

Abstract

Let ( T n ) ({T_n}) be a sequence of linear operators acting on complex valued functions on a σ \sigma -finite measure space. Assume that each T n {T_n} contracts all the p p -norms, 1 ≤ p ≤ ∞ ( i .e . ‖ T n ‖ p ≤ 1 ) 1 \leq p \leq \infty ({\text {i}}{\text {.e}}{\text {.}}{\left \| {{T_n}} \right \|_p} \leq 1) . It is shown that a.e. lim n T 1 ∗ ⋯ T n ∗ T n ⋯ T 1 f {\lim _n}T_1^ * \cdots T_n^ * {T_n} \cdots {T_1}f exists for each L p {L_p} function f , 1 > p > ∞ f,1 > p > \infty .