The Lidskii trace property and the nest approximation property in Banach spaces

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• 作者：T. Figiel^a ; ^{t.figiel@impan.gda.pl" class="auth_mail" title="E-mail the corresponding author} ; W.B. Johnson^b ; ¹ ; ^{johnson@math.tamu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 46B20 ; 46B07 ; secondary ; 46B99
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 August 2016
• 年：2016
• 卷：271
• 期：3
• 页码：566-576
• 全文大小：334 K

文摘

For a Banach space X, the Lidskii trace property is equivalent to the nest approximation property; that is, for every nuclear operator on X   that has summable eigenvalues, the trace of the operator is equal to the sum of the eigenvalues if and only if for every nest N$\mathcal{N}$ of closed subspaces of X, there is a net of finite rank operators on X  , each of which leaves invariant all subspaces in N$\mathcal{N}$, that converges uniformly to the identity on compact subsets of X  . The Volterra nest in 52b718ba6f6e20977dbdb696bab0590" title="Click to view the MathML source">L_p(0,1)$L_p(0,1)$, 1≤p<∞$1\le p<\infty$, is shown to have the Lidskii trace property. A simpler duality argument gives an easy proof of the result [2, Theorem 3.1] that an atomic Boolean subspace lattice that has only two atoms must have the strong rank one density property.