Journal of Prime Research in Mathematics

Some more remarks on grothendieck-lidskii trace formulas

Oleg Reinov
Department of Mathematics and Mechanics, St. Petersburg State University, Saint Petersburg, Russia and Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan.

\(^{1}\)Corresponding Author:


Let \(r ∈ (0, 1]\), \(1 ≤ p ≤ 2\), \(u ∈ X^∗⊗X\) and \(u\) admits a representation \(u=\sum_{i}\lambda_{i}x_{i}^{‘}⊗ x_{i}\) with \((λ_i) ∈ l_r\) bounded and \((x_{i} ∈ l^{w}_{p’} (X)\). If \(1/r + 1/2 − 1/p = 1\) then the system \(\mu_{k}\) of all eigenvalues of the corresponding operator \(\widetilde{u}\) (written according to their algebraic multiplicities) is absolutely summable and trace \(u=\sum_{k}\mu_{k}\). One of the main aim of these notes is not only to give a proof of the theorem but also to show that it could be obtained by A. Grothendieck in 1955.


 \((s, p)\)-nuclear operators, eigenvalue distributions.