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We extend a theorem of Lotz, which says that any Markov operator T acting on C(X ) such that T * is mean ergodic and all invariant measures have non-meager supports must be quasi-compact, to Lotz-Räbiger nets.

The Lotz-Räbiger nets (L R-nets) introduced in Räbiger (1993) [18] under the name M-nets provide an appropriate setting for investigation various mean ergodic theorems in Banach spaces. In the present paper we prove several convergence theorems for L R-nets of Markov operators in L 1 -spaces which extend Theorems 1 and 5 from Emel'yanov (2004) [8], and Theorem 1.1 from Lasota (1983) [11]. Preliminaries1.1. Let X be a Banach space, let L(X) be the algebra of all bounded linear operators in X , and let I = I X be the identity operator in X . A family Ψ = (T υ ) υ∈Υ ⊆ L(X) indexed by a directed set Υ = (Υ, ≺) is called an operator net. The net Ψ is strongly convergent if the norm-limit lim υ→∞ T υ x exists for each x. A vector x ∈ X is called a fixed vector for the net Ψ if T υ x = x for every υ ∈ Υ . We denote by Fix(Ψ ) the set of all fixed vectors of Ψ . It is easy to see that Fix(Ψ ) is a closed subspace of X .1.2. The following important concept had been introduced by H.P. Lotz in [16] under the name U M-sequence. Its generalization to arbitrary nets is due to F. Räbiger [18], who used the term M-nets. We prefer to use slightly modified terminology following to the recent paper [5], and call M-nets by Lotz-Räbiger nets. Definition 1. A net Ψ = (T υ ) υ∈Υ ⊆ L(X) is called a Lotz-Räbiger net if

We generalize Sine's example [3] of a positive contraction in a C(K)-space which is mean ergodic, but its square is not. (2000). 46B42, 47A35. Mathematics Subject Classification