Analogs of Weyl inequalities and the trace theorem in Banach space

Let $A$ be a completely continuous operator acting on the Banach space $\mathfrak B$, let $\{\lambda_j(A)\}$ be the complete system of its eigenvalues (with regard for multiplicity) and let $s_{n+1}(A)$ be the distance from $A$ to the set of all operators of range dimension not greater than $n$. If
\begin{equation} \sum_{n=1}^\infty s_n(A)\ln\bigl(s_n^{-1}(A)+1\bigr)<\infty, \end{equation}
then $\operatorname{sp}A=\sum\lambda_j(A)$, where $\operatorname{sp}A$ is a functional which is linear on the set of operators satisfying condition (1) (and continuous in a certain topology) and which coincides with its trace for finite-dimensional $A$. The proof of this theorem is based on certain analogs of the famous Weyl inequalities.
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