Generalization of Eberlein’s and Sine’s ergodic theorems to \(LR\)-nets. (English) Zbl 1324.47019

Emel’yanov, Eh. Yu.; Erkursun, Nazife

Vladikavkaz. Mat. Zh. 9, No. 3, 22-26 (2007).

Summary: 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 [Math. Ann. 297, No. 1, 103–116 (1993; Zbl 0788.47036)].

MSC:

47A35	Ergodic theory of linear operators
47D99	Groups and semigroups of linear operators, their generalizations and applications
47L07	Convex sets and cones of operators

Keywords:

Banach space; operator net; LR-net; strong convergence

Citations:

Zbl 0788.47036