Çakan, Celal; Altay, Bilal; Coşkun, Hüsamettin \(\sigma\)-regular matrices and a \(\sigma\)-core theorem for double sequences. (English) Zbl 1182.40003 Hacet. J. Math. Stat. 38, No. 1, 51-58 (2009). Let \(A= [a_{jk}^{mn}]_{j,k=0}^{\infty}\) be a four dimensional infinite matrix of real numbers for all \(m,n= 0,1,\dots\). The sums \[ y_{mn} = \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}a_{jk}^{mn}x_{jk} \]are called the \(A\)-transformations of the double sequence \(x\). We say that a sequence \(x\) is \(A\)-summable to the limit \(l\) if the \(A\)-transforms of \(x\) exist for all \( m,n=0,1,\dots\) and are convergent in the sense of Pringsheim, i.e., \[ \lim_{p,q} \sum_{j=0}^{p}\sum_{k=0}^{q}a_{jk}^{mn}x_{jk}=y_{mn} \]and \[ \lim_{mn}y_{mn}= l. \]Let us write \[ L^{*}(x) = \limsup_{p,q} \sup_{s,t}\frac{1}{pq}\sum_{j=0}^{p}\sum_{k=0}^{q}x_{j+s,k+t} \]and \[ C_{\sigma}(x) = \limsup_{p,q} \sup_{s,t}\frac{1}{pq}\sum_{j=0}^{p}\sum_{k=0}^{q}x_{\sigma^{j}(s)\sigma^{k}(t)}. \]In this paper the authors investigate necessary and sufficient conditions for the inequality \[ C_{\sigma}(Ax) \leq L(x) \tag{1} \]for all \(x\in l_{\infty}^{2}\). The authors also indicate that in the case \(\sigma(i) = i+1,\) the inequality in (1) reduces to \(L^{*}(Ax)\leq L(x)\). Reviewer: Ekrem Savas (Istanbul) Cited in 3 Documents MSC: 40C05 Matrix methods for summability 40B05 Multiple sequences and series Keywords:double sequences; invariant means; core theorems PDFBibTeX XMLCite \textit{C. Çakan} et al., Hacet. J. Math. Stat. 38, No. 1, 51--58 (2009; Zbl 1182.40003)