Trace-class operators and commutators
type: Item
instance of: scholarly article
title: Trace-class operators and commutators
zbMATH Open document ID: 0684.47017
DOI: 10.1016/0022-1236(89)90064-5
author: N. J. Kalton
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: If I and J are two-sided ideals in the algebra of bounded operators B on \(\ell_ 2\), denote by [I,J] the linear span of the commutators AB-BA with \(A\in I\), \(B\in J\). N. J. Kalton gives a beautiful characterization of the trace-class commutators [\({\mathcal S}_ 1,B]\), solving a problem open for some time. For the Schatten p-classes \({\mathcal S}_ p\) and \(p\neq 1\), the corresponding characterization of [\({\mathcal S}_ p,B]\) is known and simpler. For \(p=1\), one gets the linear space \({\mathcal C}{\mathcal S}_ 1\) of those \(T\in {\mathcal S}_ 1\) whose eigenvalues \(\lambda_ n(T)\) satisfy \[ \sum_{n\in {\mathbb{N}}}n^{-1}| \lambda_ 1(T)+...+\lambda_ n(T)| <\infty. \] Hence trace T\(=0\) for \(T\in {\mathcal C}{\mathcal S}_ 1\) (but not conversely). The commutator space [\({\mathcal S}_ o,{\mathcal S}_ q]\) with \(1/p+1/q=1\) also coincides with \({\mathcal C}{\mathcal S}_ 1\). At most 6 commutators are needed to reach any \(T\in {\mathcal C}{\mathcal S}_ 1\); \({\mathcal C}{\mathcal S}_ 1\) is also equal to the linear span of the quasi- nilpotent trace-class operators. The techniques used are extensios of ideas concerning nonlinear commutators arising in interpolation theory. They are also connected with the theory of twisted sums of Banach spaces due to the author.
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 47L30
zbMATH DE Number: 4120850
zbMATH Keywords: two-sided ideals in the algebra of bounded operators
zbMATH Keywords: trace-class commutators
zbMATH Keywords: Schatten p-classes
zbMATH Keywords: commutator space
zbMATH Keywords: quasi-nilpotent trace- class operators
zbMATH Keywords: twisted sums of Banach spaces
Wikidata QID: Q56689635
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90064-5
OpenAlex ID: W2068882164
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description: scientific article; zbMATH DE number 4120850