On the Spectral Theory of Operator Pencils in a Hilbert Space

Abstract

Consider the operator pencil L _λ = A − λB − λ ² C, where A, B, and C are linear, in general unbounded and nonsymmetric, operators densely defined in a Hilbert space H. Sufficient conditions for the existence of the eigenvalues of L _λ are investigated in the case when A, B and C are K-positive and K-symmetric operators in H, and a method to bracket the eigenvalues of L _λ is developed by using a variational characterization of the problem (i) L _λ u = 0. The method generates a sequence of lower and upper bounds converging to the eigenvalues of L _λ and can be considered an extension of the Temple-Lehman method to quadratic eigenvalue problems (i).