Double σ-multiplicative matrices

The class of σ-regular matrices was defined and characterized by Schaefer [P. Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1972) 104–110] and further studied by Mursaleen [Mursaleen, On some new invariant matrix methods of summability, Quart. J. Math. Oxford 34 (1983) 77–86], Ahmad and Mursaleen [Z.U. Ahmad, Mursaleen, An application of Banach limits, Proc. Amer. Math. Soc. 103 (1988) 244–246]. In this paper we characterize four-dimensional σ-multiplicative matrices, and establish a core theorem.