Operator ideals on non-commutative function spaces
type: Item
instance of: scholarly article
title: Operator ideals on non-commutative function spaces
zbMATH Open document ID: 1322.46040
author: Eugeniu Spinu
author: T. Oikhberg
published in: Integral Equations and Operator Theory
publication date: 2014-10-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1401.4231
review text: Given two Banach spaces \(X\) and \(Y\), \(B(X,Y)\) (respectively \(\mathcal{K}(X,Y)\)) denotes the set of bounded (respectively compact) linear operators from \(X\) to \(Y\). We say that \(T\in B(X,Y)\) fixes a (complemented) copy of a subspace \(Z\subset X\) if \(Z\) is isomorphic to \(T(Z)\) (with \(Z\) and \(T(Z)\) complemented in \(X\) and in \(Y\), respectively). \(T\) is called strictly singular (respectively finitely strictly singular) if it does not fix any infinite-dimensional subspace (respectively if given \(\varepsilon>0\) there is a natural number \(N\) such that each subspace of dimension at least \(N\) contains an element \(x\) with \(\|Tx\|<\varepsilon\|x\|\). The strictly singular operators form an ideal \(\mathcal{SS}(X,Y)\subset B(X,Y)\) (in the ``usual'' sense: \(\mathcal{SS}(X,Y)\) contains the finite rank operators and if \(U\in B(X_0,X)\), \(V\in B(Y,Y_0)\), then \(VTU\in\mathcal{SS}(X_0,Y_0)\)), likewise for the finitely strictly singular operators \(\mathcal{FSS}(X,Y)\subset B(X,Y)\). The following inclusions hold and are, in general, strict: \[ \begin{cases}\mathcal{K}(X,Y)\subset\mathcal{FSS}(X,Y)\subset\mathcal{SS}(X,Y)\subset\mathcal{IN}(X,Y),\\ \mathcal{K}(X,Y)\subset\mathcal{SCS}(X,Y)\subset\mathcal{IN}(X,Y), \end{cases}\tag{1} \] where \(\mathcal{IN}\) denotes the set of inessential or Fredholm perturbation operators \(T\) defined by the fact that \(I+WT\) is Fredholm for every \(W\in B(Y,X)\) and where \(\mathcal{SCS}\) denotes the strictly cosingular operators \(T\) defined by the fact that \(Q_ZT\) is not surjective, with \(Q_Z\) the quotient map \(Y\to Y/Z\). In special cases, for example, when \(X\) and \(Y\) are Hilbert spaces or \(X=Y=l_p\) (\(1\leq p<\infty\)), the inclusions above are known to coincide. The authors study the ideals above when \(X\) and \(Y\) are certain non-commutative \(L_p\)-spaces; to be more precise, they mostly consider \(L_p\)-spaces associated to a hyperfinite von Neumann algebra \(A\) with a normal f…
reviewed by: Hermann Pfitzner
Mathematics Subject Classification ID: 46L52
Mathematics Subject Classification ID: 47L20
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47B65
zbMATH DE Number: 6358019
zbMATH Keywords: non-commutative function space
zbMATH Keywords: sequence space
zbMATH Keywords: operator ideals
zbMATH Keywords: strictly singular operator
zbMATH Keywords: finitely strictly singular operator
MaRDI profile type: Publication
OpenAlex ID: W2594120075
arXiv ID: 1401.4231
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DOI: 10.1007/S00020-014-2167-4
description: scientific article; zbMATH DE number 6358019