Grothendieck-Lidskii theorem for subspaces and factor spaces of L_p-spaces

In 1955, A. Grothendieck has shown that if the linear operator \(T\) in a Banach subspace of an \(L_\infty\)-space is 2/3-nuclear then the trace of \(T\) is well defined and is equal to the sum of all eigenvalues \(\{\mu_k(T)\}\) of \(T.\) V.B. Lidski\vı, in 1959, proved his famous theorem on the co...

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Bibliographic Details
Published in:arXiv.org 2011-05
Main Authors: Reinov, Oleg, Latif, Qaisar
Format: Article
Language:English
Subjects:
Eigenvalues
Linear operators
Subspaces
Theorems
Online Access:Get full text
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Summary:In 1955, A. Grothendieck has shown that if the linear operator \(T\) in a Banach subspace of an \(L_\infty\)-space is 2/3-nuclear then the trace of \(T\) is well defined and is equal to the sum of all eigenvalues \(\{\mu_k(T)\}\) of \(T.\) V.B. Lidski\vı, in 1959, proved his famous theorem on the coincidence of the trace of the \(S_1\)-operator in \(L_2(\nu)\) with its spectral trace \(\sum_{k=1}^\infty \mu_k(T).\) We show that for \(p\in[1,\infty]\) and \(s\in (0,1]\) with \(1/s=1+|1/2-1/p|,\) and for every \(s\)-nuclear operator \(T\) in every subspace of any \(L_p(\nu)\)-space the trace of \(T\) is well defined and equals the sum of all eigenvalues of \(T.\)
ISSN:2331-8422