On grothendieck-lidskii trace formulas and applications to approximation properties

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]r∈(0,1], 1≤p≤21≤p≤2, u∈X∗|⊗r,pXu∈X∗|⊗r,pX and uu admits a representation u=∑λixi‘⊗xiu=∑λixi‘⊗xi with (λi)∈lr,(x‘i)(λi)∈lr,(xi‘) bounded and (xi)∈lwp′(X)(xi)∈lp′w(X). If 1/r+1/2−1/p=11/r+1/2−1/p=1, then the system (µk)(µk) of all eigenvalues of the corresponding operator ˜uu~ (written according to their algebraic multiplicities), is absolutely summable and trace(u)=∑µktrace(u)=∑µk.