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In an earlier paper we provided a counterexample to an old conjecture of Hopf. In this note we show that the "strong sweeping out property" obtains for the Hopf operators (Tt) both when t —> +∞ and when t —> 0+, that is a.e. convergence fails in the worst possible way.
The authors investigate which results of the classical mean ergodic theory for bounded linear operators in Banach spaces have analogues for subadditive sequences (Fn) in a Banach lattice B. A sequence (Fn) is subadditive for a positive contraction T in B if Fn+k ≤ Fn + TnFk (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−1Fn) implies In Lp (1 ≤ p < ∞) the existence of a weak cluster point for non-negative (n−1Fn) is equivalent with norm convergence. If T is an isometry in Lp (1 < p < ∞) and sup then n−1Fn converges weakly. If T in L1 has a strictly positive fixed point and sup then n−1Fn converges strongly. Most results are proved even in the d-parameter case.
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