Absolute Cesàro series spaces and matrix operators
type: Item
instance of: scholarly article
title: Absolute Cesàro series spaces and matrix operators
zbMATH Open document ID: 1396.40003
published in: Acta Applicandae Mathematicae
publication date: 2018-04-16T00:00:00Z
review text: Denote by \(l_k\), \(1\leq k<\infty\), the space of \(k\)-absolutely convergent series. Then, for \(\lambda>-1\) and \(\lambda+\mu\neq -1, -2, \dots\), the authors introduce the generalized Cesàro series space \(|C_{\lambda, \mu}|_k\) as \[ |C_{\lambda, \mu}|_k=(l_k)_T=\{x=(x_n)\in\omega: T(x)\in l_k\}, \] where \(T=(t_{n\upsilon})\) is a triangular matrix given by the following expression \[ t_{n\upsilon} = \begin{cases} 1 & \quad n, \upsilon=0,\\ \frac{A_{n-\upsilon}^{\lambda-1}{A_\upsilon^\mu}\upsilon}{n^{1/k}A_n^{\lambda+\mu}} & \quad 1\leq \upsilon \leq n, \\ 0 & \quad \upsilon>n, \end{cases} \] and \(A_0^\lambda=1\), \(A_n^\lambda=\frac{(\lambda+1)(\lambda+2)\ldots(\lambda+n)}{n!}\), \(A_{-n}^\lambda=0\), \(n\geq 1\). It is shown that the space \(|C_{\lambda, \mu}|_k\) is a \(BK\)-space induced by the norm \(\|x\|_{|C_{\lambda, \mu}|_k}=\|T(x)\|_{l_k}\) (Theorem 2.2) if \(\mu\) and \(\lambda+\mu\) are non-negative integers. The isometric characterization of the space is established in Theorem 2.3, which says \(|C_{\lambda, \mu}|_k\) is linearly isomorphic to the space \(l_k\) for \(1\leq k<\infty\). Further, the \(\beta\)-dual of \(|C_{\lambda, \mu}|_k\) and results related to matrix transformations under certain restrictions on \(\lambda, \mu\) are derived in Theorems 2.1, 2.4 and 2.5, respectively. Several consequences of the established theorems depending on the parameters \(\lambda, \mu\), etc., and the matrix \(A\) are discussed in Section 4. The results obtained in this paper strengthen several earlier known results.
reviewed by: Atanu Manna
Mathematics Subject Classification ID: 40C05
Mathematics Subject Classification ID: 40F05
Mathematics Subject Classification ID: 46A45
zbMATH DE Number: 6860441
zbMATH Keywords: absolute Cesàro summability
zbMATH Keywords: \(\beta\)-dual
zbMATH Keywords: matrix transformations
author: G. Canan Hazar Güleç
author: Mehmet Ali Sarıgöl
MaRDI profile type: Publication
full work available at URL: s10440-017-0138-x
OpenAlex ID: W2769083165
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DOI: 10.1007/S10440-017-0138-X
description: scientific article; zbMATH DE number 6860441