Grothendieck‐Lidskiǐ theorem for subspaces of L p ‐spaces

• Published in: Mathematische Nachrichten 2013-02, Vol.286 (2-3), p.279-282
• Main Authors: Reinov, Oleg, Latif, Qaisar
• Format: Article
• Language: English
• ISSN: 0025-584X; 1522-2616
• DOI: 10.1002/mana.201100112

In 1955, A. Grothendieck has shown that if the linear operator T in a Banach subspace of an L ∞ ‐space is 2/3‐nuclear then the trace of T is well defined and is equal to the sum of all eigenvalues {μ k ( T )} of T . Lidskiǐ, in 1959, proved his famous theorem on the coincidence of the trace of the S 1 ‐operator in L 2 (ν) with its spectral trace \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\sum _{k=1}^\infty \mu _k(T)$\end{document} . We show that for p ∈ [1, ∞] and s ∈ (0, 1] with 1/s = 1 + |1/2 − 1/p|, and for every s‐nuclear operator T in every subspace of any L p (ν)‐space the trace of T is well defined and equals the sum of all eigenvalues of T. Note that for p = 2 one has s = 1, and for p = ∞ one has s = 2/3.