On vector-valued Banach limits
type: Item
instance of: scholarly article
title: On vector-valued Banach limits
zbMATH Open document ID: 1326.40005
author: R. Armario
author: Francisco Javier Pérez-Fernández
author: Francisco García-Pacheco
publication date: 2015-01-05T00:00:00Z
review text: A linear functional \(\varphi: \ell^{\infty} \rightarrow \mathbb{R}\) is called a Banach limit if \(\varphi\) is positive (i.e., \(\varphi((x_n)_{n\in\mathbb{N}})\geq 0\) if \(x_n\geq 0\) for each \(n\in \mathbb{N}\)), invariant under the left-shift operator, and maps the constant sequence \((1,1,1,\dots)\) to \(1\). Every Banach limit is bounded with norm one and satisfies \(\varphi((x_n)_{n\in \mathbb{N}})=\lim_{n\to \infty}x_n\) for each convergent sequence \((x_n)_{n\in \mathbb{N}}\). The existence of Banach limits was already proved by S.\,Banach himself. In this paper, the authors give a generalisation of the concept of Banach limits to vector-valued sequences. They first note that a linear functional \(\varphi: \ell^{\infty} \rightarrow \mathbb{R}\) is a Banach limit if and only if \(\varphi\) is bounded with norm one, left-shift invariant, and satisfies \(\varphi((x_n)_{n\in \mathbb{N}})=\lim_{n\to \infty}x_n\) for every convergent sequence \((x_n)_{n\in \mathbb{N}}\). Thus they define, for a real Banach space \(X\), the set \(BL(X)\) of all vector-valued Banach limits on \(X\) as the set of all bounded linear operators \(T:\ell^{\infty}(X) \rightarrow X\) with \(\|T\|=1\) which are left-shift invariant and satisfy \(\varphi((x_n)_{n\in \mathbb{N}})=\lim_{n\to \infty}x_n\) for every norm-convergent sequence \((x_n)_{n\in \mathbb{N}}\) in \(X\). The authors show that for every real Banach space \(X\) one has \(BL(X^*)\neq \emptyset\), where \(X^*\) is the dual space of \(X\). On the other hand, for the space \(c_0\) of all real-valued null sequences (equipped with the supremum norm) one has \(BL(c_0)=\emptyset\). Also, if \(BL(X)\neq \emptyset\) and \(Y\) is a 1-complemented subspace of \(X\), then \(BL(Y)\neq \emptyset\). In [Acta Math., Uppsala 80, 167--190 (1948; Zbl 0031.29501)], \textit{G. G. Lorentz} defined a sequence \((x_n)_{n\in \mathbb{N}}\in \ell^{\infty}\) to be almost convergent to \(x\in \mathbb{R}\) if \(\varphi((x_n)_{n\in \mathbb{N}})=x\) ho…
reviewed by: Jan-David Hardtke
Mathematics Subject Classification ID: 40J05
Mathematics Subject Classification ID: 40H05
Mathematics Subject Classification ID: 46B15
zbMATH DE Number: 6383397
zbMATH Keywords: Banach limits
zbMATH Keywords: vector-valued sequences
zbMATH Keywords: almost convergence
MaRDI profile type: Publication
full work available at URL: s10688-013-0038-4
OpenAlex ID: W2017020234
cites work: Banach limits and uniform almost summability
cites work: Q4520851
cites work: Additions to the periodic decomposition theorem
cites work: A contribution to the theory of divergent sequences
cites work: Invariant Banach limits and applications
DOI: 10.1007/S10688-013-0038-4
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 6383397