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: orein51@mail.ru

### Abstract

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.

#### Keywords:

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