We consider the quadratic eigenvalue problem with selfadjoint operators R, S and T in the Hilbert space 8. The operator S is supposed to be "large" with respect to the operators R and T. For simplicity we assume that R and T have bounded inverses. If, additionally, S is uniformly positive, then (1) can be transformed into 1 -P z = (-p) S -~I ~R S -~I ~Z + ~ ,'j-'I2TS-'IZz with z = S'I2y. This eigenvalue problem has been studied in [l] and [5].
Following [1, page 1181 and [5, pages 317-3181, a natural linearization of (1) is
Since the intersection of the domains of the operators R, S and T may be non-dense, it is natural to study the problem (1) as a quadratic form problem. Replacing the operators R, S and Tby their quadratic forms r, s and t the equation ( 1) becomes
(3) p2u(y, z ) + ps(y, z ) + t(y, z) = 0 , z E g ( s ) c g ( r ) n g ( t ) .
In Subsection 2.1 we use a linearization procedure similar to the linearization used in [l] and [5] to get (2). In this way we introduce the operator A , and the indefinite scalar Supported in part by the Ministry of Science of Croatia. The research has been done while this author was visiting the Department of Mathematics, Western Washington University.