Matrix summability methods and weakly unconditionally Cauchy series
type: Item
instance of: scholarly article
title: Matrix summability methods and weakly unconditionally Cauchy series
zbMATH Open document ID: 1173.40003
DOI: 10.1216/RMJ-2009-39-2-367
author: C. Pérez-Eslava
author: J. B. Seoane-Sepúlveda
author: A. Aizpuru
published in: Rocky Mountain Journal of Mathematics
publication date: 2009-06-18T00:00:00Z
review text: Let \(X\) be a normed space and \(A=[a_{ij}]\) be an infinite matrix with real values. A sequence \(\{x_j\}\) in \(X\) has \(A\) limit \(x\) if \(\lim_i\sum _{j=1}^\infty a_{ij}x_j=x\) and we write \(A-\lim x_j=x\). A series \(\sum_jx_j\) in \(X\) is weakly unconditionally Cauchy (wuc) if \(\sum _{j=1}^\infty |x'(x_j)|<\infty\) for every \(x'\in X'\). Let \(\sum x_j\) be a (formal) series in \(X\); the authors consider the space \(S_A(\sum_jx_j)= \{\{t_j\}\in l^\infty: A-\lim\sum_{j=1}^n t_jx_j\) exists\} (with the sup-norm) and study the relationship between wuc series and the space \(S_A (\sum_jx_j)\). They show that if \(X\) is complete, \(\sup_i\sum_{j=1}^\infty |a_{ij}|=M>0\) and \(\sum_jx_j\) is wuc, then \(S_A (\sum_j x_j)\) is complete. Moreover, if \(A\) is regular and \(X\) is complete, then \(\sum_j x_j\) is wuc iff \(S_A (\sum_jx_j)\) is complete. They also consider the linear operator \(T:S_A (\sum_j x_j)\to X\) defined by \(T\{t_{j}\}=A-\lim\sum_{j=1}^{n} t_jx_j\) and show that \(T\) is continuous iff \(\sum_{j}x_{j}\) is wuc. Similar results are given for the weak analogue of the space \(S_A (\sum_j x_j)\) defined by \(S_{wA}(\sum_j x_j)= \{t_j\}\in l^\infty: \text{weak}-A-\lim \sum_{j=1}^n t_j x_j\) exists\}. Finally, the authors establish a version of the Orlicz-Pettis Theorem for series which have a weak unconditional \(A\) limit and \(A\) is a regular matrix.
Mathematics Subject Classification ID: 40C05
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46B45
zbMATH DE Number: 5566175
zbMATH Keywords: matrix summability
zbMATH Keywords: weakly unconditionally Cauchy series
zbMATH Keywords: Orlicz-Pettis theorem
reviewed by: Charles Swartz
MaRDI profile type: Publication
cites work: Unconditionally Cauchy series and Cesàro summability
cites work: Matrix methods in analysis
cites work: On bases and unconditional convergence of series in Banach spaces
cites work: Q4520851
cites work: Q3329946
cites work: On Relationships Amongst Certain Spaces Of Sequences In An Arbitrary Banach Space
cites work: Characterizations of completeness of normed spaces through weakly unconditionally Cauchy series
full work available at URL: rmj-2009-39-2-367
OpenAlex ID: W2006363103
description: scientific article; zbMATH DE number 5566175