[Authors]
 Oleg Reinov

[Title]
 Some more remarks on Grothendieck-Lidskii trace formulas

[AMS Subj-class]
 47B06 (s,p,q)-nuclear operators, eigenvalue distributions

[Abstract]
 Theorem: Let $r\in (0,1], 1\le p\le2,$ $u\in X^*\wh\otimes X$ and $u$ 
 admits a representation $$ u=\sum_i \lambda_i x'_i\otimes x_i, $$
 with $(\la_i)\in l_r,$ $(x'_i)$ bounded and  $(x_i)\in l_{p'}^w(X).$
 If $1/r+1/2-1/p=1,$ then the system $(\mu_k)$ of all eigenvalues  
 of the corresponding operator $\wt{u}$ (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.

[Comments]
 LaTeX, English, 7 pp.

[Contact e-mail]
 orein51@mail.ru