A generalization of Gerisch's theorem on biorthogonal systems
type: Item
instance of: scholarly article
zbMATH Open document ID: 0992.46009
publication date: 2002-04-24T00:00:00Z
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 40H05
zbMATH DE Number: 1733181
zbMATH Keywords: biorthogonal system
zbMATH Keywords: dual pair
zbMATH Keywords: summability
zbMATH Keywords: linear summing method
zbMATH Keywords: Gerisch's theorem
author: Vladimir Strauss
MaRDI profile type: Publication
title: A generalization of Gerisch's theorem on biorthogonal systems
published in: Methods of Functional Analysis and Topology
review text: \textit{W. Gerisch} [Dokl. Akad. Nauk SSSR 200, 1018-1019 (1971; Zbl 0235.42008)] considered a biorthogonal pair of sequences \(\{ e_i\}\) and \(\{ g_i\}\) in a Hilbert space, each of them being complete, and a numerical sequence \(\{ \varepsilon_i=\pm 1\}\) such that the series \(\sum \varepsilon_i|(x,e_i)|^2\) and \(\sum \varepsilon_i|(x,g_i)|^2\) converge for every vector \(x\). It was proved that each of the above sequences of vectors is a basis (possibly conditional). NEWLINENEWLINENEWLINEThe author gives the following generalization: the biorthogonal sequences are considered in a dual pair of Banach spaces; instead of the convergence, the series are summable by certain linear methods, the basis property is replaced by an appropriate summability assertion. Some examples are given.
reviewed by: Anatoly N. Kochubei
description: scientific article; zbMATH DE number 1733181