Local ergodic theorems for N-parameter semigroups of operators

Abstract

The main purpose of this paper is to prove the following local ergodic theorem. If {T(t₁,…,t_N): t_i ≥ 0, i=1,…,N} is an N-parameter strongly continuous semigroup of positive contraction operators on L₁(Θ,F,μ), then for all f є L₁(Θ) $M\left( \varepsilon \right)f = \frac{1}{{_\varepsilon N}}\int_o^\varepsilon \ldots \int_o^\varepsilon {T\left( {t_1 , \ldots ,t_N } \right)f dt_1 } \ldots dt_N$ converges almost everywhere as ε + 0 + . The theorem will also be proved for semigroups of non-positive L₁ contractions which are also L_∞ contracting. Examples are given to show that the maximal ergodic lemmas which are used to prove the one parameter case of the above results do not extend to the N-parameter case. The N-parameter case is proved by successive reduction of the number of parameters.

The results of this paper are to appear in the author's Ph.D. thesis, which is being written under the direction of Professor U. Krengel.

Part of this work has been supported through N.S.F. grant GP 9354.