On the closed subideals of \(L(\ell _{p} \oplus \ell _{q})\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1267.47111
author: Th. Schlumprecht
published in: Operators and Matrices
publication date: 2012-09-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.3610
Mathematics Subject Classification ID: 47L20
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 6082489
zbMATH Keywords: operator ideals
zbMATH Keywords: \(\ell_p\)-spaces
zbMATH Keywords: closed subideals
zbMATH Keywords: finitely strictly singular operators
MaRDI profile type: Publication
OpenAlex ID: W2320946465
arXiv ID: 1105.3610
DOI: 10.7153/OAM-06-22
Recommended article: On norm closed ideals in L(lp,lq)
Recommended article: Some results on the lattice of closed ideals of $\mathcal L^r(X)$ for $X$ of the form $(\bigoplus_i \ell _p^i)_q$
Recommended article: On Closed Subspaces of ω ∗
Recommended article: On quasi \({\mathcal H}\)-closed subspaces
Recommended article: Ideaux fermés de $L^1(\mathsf{R}_+)$.
Recommended article: Q4240369
Recommended article: On \(S\)-closed subspaces
Recommended article: On \(s\)-closed subspaces
Recommended article: The closedness of shift invariant subspaces in \(L^{p,q}(\mathbb{R}^{d+1})\)
Recommended article: On the closed subfields of \(\overset\simeq {\mathbb{Q}}_p\)
title: On the closed subideals of \(L(\ell _{p} \oplus \ell _{q})\)
review text: In this paper, the author first reviews the known results about closed subideals of the space of bounded operators on \(\ell_p \otimes \ell_q\), \(1 <p<p<\infty\). It turns out that, apart from two explicitly stated maximal proper closed subideals (induced by the identity maps \(I_{\ell_p}\) and \(I_{\ell_q}\), respectively), all other closed subideals of \(L(\ell_p \otimes \ell_q)\) can be identified with the closed subideals of \(L(\ell_p,\ell_q)\). The author then exhibits two new closed and incomparable subideals of \(L(\ell_p,\ell_q)\) in the case \(1<p<2<q<\infty\), which increases the count of the closed proper and nontrivial subideals of \(L(\ell_p,\ell_q)\) to \(7\). These two ideals are those which are induced by the formal inclusions \(I(p,2): \ell_p \rightarrow \ell_2\) and \(I(2,q): \ell_2 \rightarrow \ell_q\), respectively.
reviewed by: Carsten Michels
description: scientific article; zbMATH DE number 6082489