On the completeness of derived chains

We study the problem of completeness of the system of eigenvectors and associated vectors of operator-valued functions which are analytic in an angular region and which assume values in the ring $\mathfrak R$ of bounded linear operators in a separable Hilbert space $\mathfrak H$. As a corollary of the fundamental theorem proved in this paper we obtain the following result.
Theorem 1. {\it Let $L(\lambda)=I-B_0H^\beta-\lambda B_1 H^{1+\beta}-\dots-\lambda^{n-1}B_{n-1}H^{n-1+\beta}-\lambda^nH^n,$ where $\beta>0$. $B_k\in\mathfrak R$ and $H$ is a completely continuous positive operator, moreover, let $\varliminf us^q_u(H)=0$ for some $q>0$. Then for every $\varepsilon>0$ the closed linear hull of the eigenvectors and associated vectors of $L(\lambda)$ (or $L^*(\overline\lambda)$) which correspond to the eigenvalues lying in the angular region $|\arg\lambda|<\varepsilon$ has finite defect in $\mathfrak H$.}
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