Journal of Prime Research in Mathematics

On grothendieck-lidskii trace formulas and applications to approximation properties

Qaisar Latif
Jacobs University Bremen, Research 1, Bremen Deutschland.
Abdus Salam School of Mathematical Sciences GC University Lahore, 68-B New Muslim
Town, 54600 Lahore Pakistan.

$$^{1}$$Corresponding Author: lubayeifr@gmail.com.

Abstract

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 $$trace(u) =\sum µ_k$$.

Keywords:

Eigenvalue distributions, approximation properties, trace formulas, r-nuclear operators.