Subadditive mean ergodic theorems

Abstract

Abstract The authors investigate which results of the classical mean ergodic theory for bounded linear operators in Banach spaces have analogues for subadditive sequences ( F n ) in a Banach lattice B . A sequence ( F n ) is subadditive for a positive contraction T in B if F n + k ≤ F n + T n F k ( n , k ≥ 1). For example, von Neumann's mean ergodic theorem fails to extend to the general subadditive case, but it extends to the non-negative subadditive case. It is shown that the existence of a weak cluster point f = Tf for ( n −1 F n ) implies In L p (1 ≤ p < ∞) the existence of a weak cluster point for non-negative ( n −1 F n ) is equivalent with norm convergence. If T is an isometry in L p (1 < p < ∞) and sup then n −1 F n converges weakly. If T in L 1 has a strictly positive fixed point and sup then n −1 F n converges strongly. Most results are proved even in the d -parameter case.

Locations

• Ergodic Theory and Dynamical Systems - View - PDF