Completeness of root vectors of a Keldysh pencil perturbed by an analytic operator-valued function $S(\lambda)$ with $S(\infty)=0$

The multiple completeness of the root vectors of the pencil
$$L(\lambda)=I-T_0-\lambda T_1H-\dots-\lambda^{n-1}T_{n-1}H^{n-1}-\lambda^nH^n-S(\lambda),$$
where $I$ is the identity operator in the separable Hilbert space $\mathfrak H$, $S(\lambda)$ is an operator-valued function analytic for $|\lambda|>\eta$ with $S(\infty)=0$, and $T_k$ and $H$ are completely continuous operators, is studied. The method suggested in this note for proving the completeness does not use the factorization theorems, due to which we can remove certain restrictions on the function $S(\lambda)$ connected with the application of the factorization theorems.