On grothendieck-lidskii trace formulas and applications to approximation properties

The purpose of this short note is to consider the questions in connection with famous the Grothendieck-Lidskii trace formulas, to give an alternate proof of the main theorem from [10] and to show some of its applications to approximation properties:
Theorem: Let \(r ∈ (0, 1]\), \(1 ≤ p ≤ 2\), \(u ∈ X^{∗}|⊗_{r,p}X\) and \(u\) admits a representation \(u=\sum \lambda_{i}x_{i}{‘} ⊗x_{i}\) with \((λi) ∈ l_r,(x_{i}^{‘})\) bounded and \((x_i) ∈ l_{p’}^{w} (X)\). If \(1/r + 1/2 − 1/p = 1\), then the system \((µ_k)\) of all eigenvalues of the corresponding operator \(\widetilde{u}\) (written according to their algebraic multiplicities), is absolutely summable and \(trace(u) =\sum µ_k\).