Spectral Properties of Some Operator, Rationally Depending on Parameter

It is investigated the spectral properties of some operator, rationally depending on parameter in Hilbert space. It is known that the problem of oscillations of viscous liquid located in a stationary vessel and having a free surface, leads to the investigation of such type operator equations. Methods of investigations in this work are the methods of functional analysis, spectral theory of operators, theory of functions and multiparameter system of operators. It is famous that the notions of completeness of eigen and associated (e.a.) vectors, multiple completeness of them, question of existence of bases of e.a. vectors, multiple bases, asymptotic of eigenvalues are the fundamental directions in spectral theory of operators in Hilbert space. In this paper the authors are proved the possibility of existence of two bases of e.a. vectors of considerable equation in the Hilbert space, proved the existence of two sequences of eigenvalues of operator equation and the asymptotic of corresponding eigenvalues.