Some more remarks on grothendieck-lidskii trace formulas

Abstract

Let r∈(0,1]r∈(0,1], 1≤p≤21≤p≤2, u∈X∗⊗Xu∈X∗⊗X and uu admits a representation u=∑iλix‘i⊗xiu=∑iλixi‘⊗xi with (λi)∈lr(λi)∈lr 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 μkμk of all eigenvalues of the corresponding operator ˜uu~ (written according to their algebraic multiplicities) is absolutely summable and trace u=∑kμku=∑kμk. 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.