On some generalizations of Lorentzʹs almost convergence

We investigate an extension of the almost convergence of G.G. Lorentz, further weakening the notion of M-almost convergence we defined in [S. Mercourakis, G. Vassiliadis, An extension of Lorentzʹs almost convergence and applications in Banach spaces, Serdica Math. J. 32 (2006) 71–98] and requiring that the means of a bounded sequence restricted on a subset M of N converge weakly in ℓ ∞ ( M ) . The case when M has density 1 is of special interest and in this case we derive a result in the direction of the Mean Ergodic Theorem (see Theorem 2).