Frames and Riesz bases for Banach spaces, and Banach spaces of vector-valued sequences
type: Item
instance of: scholarly article
title: Frames and Riesz bases for Banach spaces, and Banach spaces of vector-valued sequences
zbMATH Open document ID: 1271.46013
author: Kyugeun Cho
author: Ju Myung Kim
author: Han Ju Lee
published in: Banach Journal of Mathematical Analysis
publication date: 2013-03-20T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bjma/1363784230
review text: The authors give some relationships between Bessel sequences (respectively, frames) and Riesz sequences (respectively, Riesz bases). Atomic decomposition and Banach frames in Banach spaces are also discussed. Finally, Banach spaces consisting of Bessel or Riesz sequences are introduced and it is proved that they are isometrically isomorphic to some Banach spaces of bounded linear operators, and that some subspaces of those Banach spaces are isometrically isomorphic to some Banach spaces of compact operators. The result given in Proposition \(4.1\), which provides necessary and sufficient conditions for the existence of a Banach frame in a Banach space, was already proved by \textit{P. K. Jain}, \textit{S. K. Kaushik} and \textit{L. K. Vashisht} [Indian J. Pure Appl. Math. 37, No. 5, 265-272 (2006; Zbl 1125.46013)]. Furthermore, \textit{P. G. Casazza} et al.[Contemp. Math. 247, 149--182 (1999; Zbl 0947.46010)] proved that every separable Banach space has a Banach frame. This result was further improved by Jain et al., where it is proved that for the existence of a Banach frame in a Banach space \(X\), separability of \(X\) is not required.
reviewed by: Lalit Kumar Vashisht
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 46B45
zbMATH DE Number: 6147466
zbMATH Keywords: Banach sequence space
zbMATH Keywords: frame
zbMATH Keywords: Riesz basis
zbMATH Keywords: atomic decomposition
zbMATH Keywords: approximation property
MaRDI profile type: Publication
OpenAlex ID: W2171749255
cites work: Q5317928
cites work: \(p\)-frames and shift invariant subspaces of \(L^p\)
cites work: Duality, reflexivity and atomic decompositions in Banach spaces
cites work: The reconstruction formula for Banach frames and duality
cites work: Approximation properties.
cites work: The art of frame theory
cites work: Frame expansions in separable Banach spaces
cites work: Q4945742
cites work: An introduction to frames and Riesz bases
cites work: Perturbations of Banach Frames and Atomic Decompositions
cites work: \(p\)-frames in separable Banach spaces
cites work: The Approximation Problem for Banach Spaces
cites work: Q3518601
cites work: Describing functions: Atomic decompositions versus frames
cites work: Q4136031
cites work: An Introduction to Banach Space Theory
cites work: Q2776774
cites work: Onp-frames and reconstruction series in separable Banach spaces
DOI: 10.15352/BJMA/1363784230
description: scientific article; zbMATH DE number 6147466