[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