Weak trace-class operator: Difference between revisions

Revision as of 14:41, 25 December 2016

Revision as of 14:42, 25 December 2016

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A [[compact operator]] ''A'' on an infinite dimensional [[separable space|separable]] [[Hilbert space]] ''H'' is ''weak trace class'' if μ(''n'',''A'') {{=}} O(''n''⁻¹), where μ(''A'') is the sequence of [[singular value]]s. In mathematical notation the two-sided [[ideal (ring theory)|ideal]] of all weak trace-class operators is denoted,

::::<math> L_{1,\infty} = \{ A \in K(H) : \mu(n,A) = O(n^{-1}) \}. </math>

where <math>K(H) </math> are the compact operators.{{what|reason= This definition disagrees with the definition in the article abouton the [[Dixmier trace]].}} The term weak trace-class, or weak-''L''₁, is used because the operator ideal corresponds, in J. W. Calkin's [[Calkin correspondence|correspondence]] between two-sided ideals of bounded linear operators and rearrangement invariant sequence spaces, to the [[Lp space|weak-''l''₁ sequence space]].

== Properties ==