The set of almost convergent sequences as intersections of summability fields
Given an infinite matrix \(A=(a_{nk})\) with \(a_{nk}\) numbers (real or complex), we denote by \(C_ A\) the class of all sequences of numbers \(\{x_ k\}\) such that the sequence \(\{X_ k\}\), with \(X_ k=\sum^{\infty}_{r=0}a_{kr}x_ r\), is convergent. It is known that the class of all almost convergent sequences \textbf{ac} can be expressed as the intersection of the classes: \(C_ A\) with A in the class \({\mathcal C}\) of generalized Cesàro matrices [see \textit{G. M. Petersen}, Quart J. Math., Oxford II. Ser. 7, 188-191 (1956; Zbl 0072.283)]. The author improves this result by expressing \textbf{ac} as the intersection of \(C_ A\) with A in a subclass of \({\mathcal C}\). \textit{F. Moricz} and \textit{B. E. Rhoades} [Math. Proc. Camb. Philos. Soc. 104, 283-294 (1988; Zbl 0675.40004), see also \textit{B. E. Rhoades} and \textit{X. Shi}, Res. Notes Math. 133, 173-176 (1985; Zbl 0579.40004)] have conjectured that \textbf{ac} should also be expressible as the intersection of \(C_ A\) with A in a certain specialised regular class of matrices. Following \textit{J. A. Siddiqi} [Pac. J. Math. 39, 235-251 (1971; Zbl 0229.42005), proof of Th. 5] the author provides an example of an infinite matrix A, with \(a_{nk}\) as complex numbers, which leads to show that the foregoing conjecture is not true, in general. The reviewer feels that it may be worthwhile to examine if the conjecture is valid when \(a_{nk}\) are required to be real numbers.
scientific article; zbMATH DE number 4190330
scientific article