Multiple completeness of root vectors of a Keldysh pencil perturbed by an operator-valued function analytic in a disc

In this paper a formulation of the problem of multiple completeness is considered that generalizes the earlier well-known formulation of this problem in differential equations, and also in mechanics. To within a finite-dimensional subspace, multiple completeness tests are given for a Keldysh pencil perturbed by an analytic operator-valued function. In several cases the domain of multiple completeness is found.
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