Operator ranges and spaceability
type: Item
instance of: scholarly article
title: Operator ranges and spaceability
zbMATH Open document ID: 1219.46005
author: Derek Kitson
author: Richard M. Timoney
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-03-29T00:00:00Z
full work available at URL: http://hdl.handle.net/2262/54422
review text: It is well known [\textit{A.~Plichko,} Funct. Anal. Appl, 15, 67--68 (1981; Zbl 0535.46008)] that, if for a continuous linear operator \(T\) from a Banach space \(X\) to a Banach space \(Y\) we have \(\dim Y/TX=\infty\), then \(Y\) contains a closed infinite-dimensional subspace \(Z\) such that \(Z\cap TX=\{0\}\). A similar fact is valid for Fréchet spaces [\textit{L.~Drewnowski}, Stud. Math. 77, 373--391 (1984; Zbl 0552.46003)]. In the paper under review, the authors prove some results of this kind. In particular, they prove that, if \(X_n\) are Banach spaces, \(Y\) is a Fréchet space, \(T_n:X_n\to Y\) are continuous linear operators and the subspace \(Y':=\text{lin}(T_nX_n)_1^{\infty}\subset Y\) is not closed, then \(Y\) contains a closed infinite-dimensional subspace \(Z\) with \(Z\cap Y'=\{0\}\). The authors ask if this result can be extended to allow the \(X_n\) to be Fréchet spaces.
reviewed by: A. M. Plichko
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46B03
zbMATH DE Number: 5871224
zbMATH Keywords: absolutely summing operator
zbMATH Keywords: ascent
zbMATH Keywords: descent
zbMATH Keywords: lineable
zbMATH Keywords: spaceable
MaRDI profile type: Publication
OpenAlex ID: W2066145712
cites work: Q4793616
cites work: Lineability and spaceability of sets of functions on $\mathbb {R}$
cites work: Linearity of sets of strange functions
cites work: Q3276081
cites work: Lineability of the set of bounded linear non-absolutely summing operators
cites work: Cotype and absolutely summing linear operators
cites work: Compact, non-nuclear operators
cites work: Étude sur les variétés et les opérateurs de Julia, avec quelques applications
cites work: Sur les variétés \(J\) d'un espace de Hilbert
cites work: Quasi-complements in F-spaces
cites work: On operator ranges
cites work: Q4488051
cites work: Schlichte Potenzreihen, die auf \(\mid z\mid=1\) gleichmäßig, aber nicht absolut konvergieren
cites work: On lineability of sets of continuous functions
cites work: Q3260686
cites work: Basic sequences in F-spaces and their applications
cites work: The basic sequence problem
cites work: Q4080169
cites work: Ascent and descent for sets of operators
cites work: Some Theorems from Geometric Function Theory: Applications
cites work: Bounded linear non-absolutely summing operators
cites work: Every Countable-Codimensional Subspace of a Barrelled Space is Barrelled
cites work: Q4257161
cites work: Semifredholm maps of FK spaces
cites work: Real Proofs of Complex Theorems (and Vice Versa)
DOI: 10.1016/J.JMAA.2010.12.061
description: scientific article; zbMATH DE number 5871224