J. Math. Anal. Appl. 327 (2007) 991–996 www.elsevier.com/locate/jmaa Double σ -multiplicative matrices Mursaleen a , S.A. Mohiuddine b,∗ a Department of Mathematics, Faculty of Science, King Abdul Aziz University, PO Box 80203, Jeddah, Saudi Arabia b Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India Received 7 September 2005 Available online 5 June 2006 Submitted by William F. Ames Abstract 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 [Mur- saleen, 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 estab- lish a core theorem.  2006 Elsevier Inc. All rights reserved. Keywords: Double sequences; P -Convergence; σ -Regular matrices 1. Introduction and background Let l ∞ and c be the Banach spaces of bounded and convergent sequences respectively with the sup-norm. Let σ be a one-to-one mapping from the set N of natural numbers into itself. A continuous linear functional ϕ on l ∞ is said to be an invariant mean or a σ -mean if and only if (i) ϕ(x)  0 when the sequence x = (x k ) has x k  0 for all k , (ii) ϕ(e) = 1, where e = (1, 1, 1,...), and (iii) ϕ(x) = ϕ((x σ(k) )) for all x ∈ l ∞ . Throughout this paper we consider the mapping σ which has no finite orbits, that is, σ p (k) = k for all integer k  0 and p  1, where σ p (k) denotes the pth iterate of σ at k . Note that, a σ -mean extends the limit functional on c in the sense that ϕ(x) = lim x for all x ∈ c (see [5]). Conse- * Corresponding author. E-mail addresses: mursaleenm@gmail.com (Mursaleen), mohiuddine@math.com (S.A. Mohiuddine). 0022-247X/$ – see front matter  2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.04.081