Matrix transformations between the spaces of Cesàro sequences and invariant means

Let ω be the space of all sequences, real or complex, and let l∞ and c, respectively, be the Banach spaces of bounded and convergent sequences x = (xn) with norm ‖x‖ = supk≥0 |xk|. Let σ be a mapping of the set of positive integers into itself. A continuous linear functional φ on l∞ is said to be an invariant mean or a σmean if and only if (i) φ(x) ≥ 0, when the sequence x = (xn) has xn ≥ 0 for each n; (ii) φ(e) = 1, where e = (1,1,1, . . .); and (iii) φ((xσ(n)))= φ(x), x ∈ l∞. For certain kinds of mappings, every σmean extends the limit functional φ on c in the sense that φ(x) = limx for x ∈ c (see [2, 15]). Consequently, c ⊂ cσ , where cσ is the set of bounded sequences, all of whose invariant means are equal (see [1, 9, 10]). When σ is translation, the σmeans are classical Banach limits on l∞ (see [2]) and cσ is the set of almost convergent sequences ĉ (see [7]). Almost convergence for double sequences was introduced and studied by Móricz and Rhoades [8] and further by Mursaleen and Savaş [13], Mursaleen and Edely [12], and Mursaleen [11]. If x = (xn), write Tx = (Txn)= (xσ(n)), then