On the eigenvalue problem Tu -λ Su =0 with unbounded and nonsymetric operators T and S

In this paper we study both the theoretical problem of the existence and the practical problem of the approximate calculation of eigenvalues and eigenvectors of (i) = 0, where T and S are some linear (in general unbounded and nonhermitian) operators in a Hilbert space. After a short discussion of a class of K -symmetric operators, in section 2 the author proves the existence of eigenvalues and eigenvectors of (i) under various conditions on T and S and investigates conditions under which the set of eigenvectors of (i) is complete. Section 3 indicates briefly the applicability and the unifying property of the generalized method of moments to the approximate solution of (i). Section 4 presents and thoroughly studies a very general new iterative method for the approximate solution of (i). The advantage of this method is that it does not require the practically inconvenient preliminary reduction of (i) to an equivalent problem with bounded operators and that under certain rather general conditions the convergence is mono tonic. Furthermore, by specializing operators and parameters, our iterative method contains as a special case, almost every known iterative method for the calculation of eigenvalues (mostly proved previously only for symmetric matrices and bounded operators). Finally the applicability and the numerical effectiveness of the iterative method is illustrated by calculating the smallest eigenvalue for a selfadjoint and nonselfadjoint eigenvalue problems arising in the problems of elastic stability.