Geodesic
Multiple completeness in the sense of M.\,V.~Keldysh of root functions of elliptic boundary value problems with a~polynomial spectral parameter
Izvestiya. Mathematics , Tome 28 (1987) no. 2, pp. 413-420

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In spite of the fact that the theory of solvability of elliptic boundary value problems with a polynomial spectral parameter was completed in the 1960s, up till now multiple completeness of the root functions of such problems has not been proved. In this paper the author first establishes multiple completeness of the root vectors of a polynomial unbounded operator pencil that is not of Keldysh form, and then as an application the indicated gap is filled. Bibliography: 13 titles. 
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     author = {S. Ya. Yakubov},
     title = {Multiple completeness in the sense of {M.\,V.~Keldysh} of root functions of elliptic boundary value problems with a~polynomial spectral parameter},
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S. Ya. Yakubov. Multiple completeness in the sense of M.\,V.~Keldysh of root functions of elliptic boundary value problems with a~polynomial spectral parameter. Izvestiya. Mathematics , Tome 28 (1987) no. 2, pp. 413-420. http://geodesic.mathdoc.fr/item/IM2_1987_28_2_a7/