Weakly mixing properties of semigroups of linear operators

Abstract

The concepts of strong and weak mixing play an important role in the theory of measure-preserving transformations.The strongly mixing condition was connected with the mean ergodic theorem by Blum and Hanson [3], Brunei and Keane [4], and Hanson and Pledger [7], and was generalized to transformations in infinite measure spaces by Krengel and Sucheston [17].The strongly mixing properties of linear operators of Lj-spaces have been investigated by Lin [18], Akcoglu and Sucheston [1, 2], Sato [21], and Fong and Sucheston [6].On the other hand, the weakly mixing properties have been generalized to linear operators on general Banach spaces in connection with the mean ergodic theorem by Jones [11,12,13], Nagel [20], and Jones and Lin [14].Let T be a linear operator on a Banach space E. A typical condition meaning strong mixing of T is stated as follows : for each x^E, T n x converges weakly.A corresponding condition of weak mixing is as follows: for each x<^E, there exists a subsequence {n k } of density 1 such that T nk x converges weakly.In this paper, we shall consider the weakly mixing properties of discrete cyclic semigroups and one parameter semigroups of linear operators on Banach spaces.§ 1 contains some preliminaries concerning upper and lower densities.In § 2 we shall present results concerning the weakly mixing properties of semigroups on Banach spaces.We shall introduce several conditions meaning the weakly mixing properties and including the conditions given in [11], [13], [14], and [20].Among those conditions, we shall obtain a number of implications.In § 3 we shall give further results for operator convergence of weak mixing type.In §4 we shall consider semigroups of positive linear operators on ALspaces and strengthen theorems in § 3.The author would like to express his hearty thanks to Professor H. Umegaki for his constant encouragement and valuable suggestions.§ 1. Preliminaries.Let / be a subset of the positive integers Z + -{1, 2, •••},

Locations

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