Geodesic
Generalization of Eberlein's and Sine's ergodic theorems to $LR$-nets
Vladikavkazskij matematičeskij žurnal, Tome 9 (2007) no. 3, pp. 22-26 Cet article a éte moissonné depuis la source Math-Net.Ru

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The notion of $LR$-nets provides an appropriate setting for study of various ergodic theorems in Banach spaces. In the present paper, we prove Theorems 2.1, 3.1 which extend Eberlein's and Sine's ergodic theorems to $LR$-nets. Together with Theorem 1.1, these two theorems form the necessary background for further investigation of strongly convergent $LR$-nets. Theorem 2.1 is due to F. Räbiger, and was announced without a proof in [1]. 
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E. Yu. Emel'yanov; N. Erkursun. Generalization of Eberlein's and Sine's ergodic theorems to $LR$-nets. Vladikavkazskij matematičeskij žurnal, Tome 9 (2007) no. 3, pp. 22-26. http://geodesic.mathdoc.fr/item/VMJ_2007_9_3_a2/

[1] Räbiger F., “Stability and ergodicity of dominated semigroups. II: The strong case”, Math. Ann., 297 (1993), 103–116 | DOI | MR | Zbl

[2] Lotz H. P., “Tauberian theorems for operators on Banach spaces”, Semesterbericht Functionalanalysis, WS-1983/84, Tübingen, 1–15

[3] Krengel U., Ergodic Theorems, De Gruyter, Berlin–New York, 1985 | MR | Zbl