Dichotomy theorems for random matrices and closed ideals of operators on \((\bigoplus _{n=1}^{\infty }\ell _{1}^{n})_{c_o}\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1255.46007
author: Niels Jakob Laustsen
author: Th. Schlumprecht
author: A. Zsák
author: E. Odell
publication date: 2012-08-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.2923
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 46H35
Mathematics Subject Classification ID: 46B09
Mathematics Subject Classification ID: 47L20
Mathematics Subject Classification ID: 60B20
zbMATH DE Number: 6073817
zbMATH Keywords: dichotomy theorems
zbMATH Keywords: closed ideals of operators
zbMATH Keywords: random matrices
MaRDI profile type: Publication
OpenAlex ID: W2151296166
arXiv ID: 1009.2923
DOI: 10.1112/JLMS/JDR083
title: Dichotomy theorems for random matrices and closed ideals of operators on \((\bigoplus _{n=1}^{\infty }\ell _{1}^{n})_{c_o}\)
published in: Journal of the London Mathematical Society. Second Series
review text: The authors prove two dichotomy theorems about sequences of operators into \(L_{1}\) given by random matrices. These results provide information on the closed ideal structure of the Banach algebra of all operators on the space \(\left( \bigoplus_{n=1}^{\infty}\ell_{1}^{n}\right) _{c_{0}}.\) One of the main results of the paper is the following theorem:NEWLINENEWLINEFor each positive integer \(m\), let \(T^{(m)}:\ell_{\infty}^{m}\left( \ell _{1}^{m}\right) \rightarrow L_{1}\) be an operator such that the entries of the corresponding random matrix \(\left( T_{i,j}^{(m)}\right) \) form a sequence of independent, symmetric random variables with NEWLINE\[NEWLINE \left\| T^{(m)}\right\| =\max\left\{ \mathbb{E}\left|\sum_{i=1}^{m} T_{i,j_{i}}^{(m)}\right| :j_{1},\dots,j_{m}\in\{1,\dots,m\} \right\} \leq1. NEWLINE\]NEWLINE ThenNEWLINENEWLINE(i) either the identity operators \(Id_{\ell_{1}^{k}}\) (\(k\in\mathbb{N}\)) uniformly factor through the \(T^{(m)},\)NEWLINENEWLINE(ii) or the \(T^{(m)}\) uniformly approximately factor through \(\ell_{\infty }^{k}\) (\(k\in\mathbb{N}\)).
reviewed by: Daniel Pellegrino
description: scientific article; zbMATH DE number 6073817