Summability and duality
type: Item
instance of: scholarly article
title: Summability and duality
author: Soumitra Ghara
author: Javad Mashreghi
author: Thomas J. Ransford
published in: Publicacions Matemàtiques
publication date: 2024-08-20T00:00:00Z
review text: Let \(X\) be a Banach space over the field of real or complex numbers, and let \(X^{\ast}\) be its dual. Let \(\left( e_{k}\right) _{k\geq0}\subset X\) and \(\left( \psi_{i}\right) _{i\geq0}\subset X^{\ast}\) be sequences such that \(\left\langle e_{k},\psi_{i}\right\rangle \neq0\) if and only if \(k=i.\) Denote by \(Z\) the norm closure of span\(\left\{ \psi_{k}\mid k\geq0\right\} \) and \(c_{k}:=\left\langle e_{k},\psi_{k}\right\rangle ^{-1}.\)\N\NFor every \(k,\) define the operator \(P_{k}\colon X\rightarrow X\) by \(P_{k}\left( x\right) :=c_{k}\left\langle x,\psi_{k}\right\rangle e_{k}\); then \(\left\Vert P_{k}\right\Vert =\left\vert c_{k}\right\vert \left\Vert e_{k}\right\Vert \left\Vert \psi_{k}\right\Vert \) and the adjoint operator \(P_{k}^{\ast}\colon X^{\ast}\rightarrow X^{\ast}\) is given by \(P_{k}^{\ast }\left( \phi\right) :=c_{k}\left\langle \phi,e_{k}\right\rangle \psi_{k}.\)\N\NThe authors of the paper under review consider summability matrices \(A=\left( a_{nk}\right) _{n,k\geq0}\) with \N\[\N\sum_{k=0}^{\infty}\left\vert a_{nk}\right\vert \left\Vert P_{k}\right\Vert <\infty\tag{1}\N\]\Nand define\N\[\NS_{n}^{A}\left( x\right) :=\sum_{k=0}^{\infty}a_{nk}P_{k}\left( x\right) \text{ \ }\left( x\in X\right)\N\]\Nand\N\[\N\left( S_{n}^{A}\right) ^{\ast}\left( \phi\right) :=\sum_{k=0}^{\infty }a_{nk}P_{k}^{\ast}\left( \phi\right) \text{ \ }\left( \phi\in X^{\ast }\right)\N\]\Nfor each \(n.\) The following main results are proved.\N\NTheorem 3.1 (Equivalence theorem). If there exists a matrix \(A^{0}\) (satisfying (1)) such that \(S_{n}^{A^{0}}\left( x\right) \rightarrow x\) in \(X\) for each \(x\in X,\) then the following statements are equivalent for every \(A\) with (1):\N\begin{itemize}\N\item[(i)] \N\(S_{n}^{A}\left( x\right) \rightarrow x\) weakly in \(X\) for each \(x\in X;\)\N\item[(ii)] \N\(S_{n}^{A}\left( x\right) \rightarrow x\) in \(X\) for each \(x\in X;\)\N\item[(iii)]\N\(\left( S_{n}^{A}\right) ^{\ast}\left( \phi\right) \rightarrow\phi\) …
reviewed by: Toivo Leiger
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 40H05
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 30H10
Mathematics Subject Classification ID: 30H20
Mathematics Subject Classification ID: 30H45
zbMATH DE Number: 7898906
zbMATH Keywords: Banach space
zbMATH Keywords: Cesàro mean
zbMATH Keywords: dual space
zbMATH Keywords: limitation theorem
zbMATH Keywords: summability
MaRDI profile type: Publication
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DOI: 10.5565/PUBLMAT6822403
description: scientific article; zbMATH DE number 7898906