On the asymptotics of eigenvalues of an analytical pencil of operators (Q1873219)

This paper is concerned with the asymptotic behavior of eigenvalues and eigenvectors of the perturbed pencil \(\mathcal{A}_\epsilon (\lambda)=\mathcal{A}(\lambda)+ \epsilon \mathcal{B}(\lambda)\), \(\lambda \in \mathbb{C},\) under the assumption that \(\mathcal{A} (\lambda) = \sum_{k=O}^{\infty}A_{k} \lambda^{k} \) and \(\mathcal{B} (\lambda) =\sum_{k=0}^{\infty} B_{k} \lambda^{k} \) are analytical pencils of operators from a Banach space \(E_1\) to a Banach space \(E_2,\) where \(A_0\) and \(B_0\) are Fredholm operators, \(A_k\) and \(B_k,\) \(k\geq 1,\) are compact operators and satisfy \(||A_k||<C_A R_A^k \) and \(||B_k||<C_B R_B^k.\) The main results are stated under the conditions that \(\lambda_0 =0 \) is an isolated eigenvalue of the pencil \(\mathcal{A} (\lambda)\) and that for every \(\lambda\) such that \(0< |\lambda|\leq r_o\) there exists a bounded inverse operator \(\mathcal{A} (\lambda)^{-1}.\) The author shows, in particular, that, if for \(\lambda _0\), \(A_0\) has a complete system of Jordan chains having \(N\) elements, then, for \(\epsilon \) sufficiently small, \(\lambda _0\) splits into \(N\) eigenvalues of \(\mathcal{A}_{\epsilon}(\lambda) \) which can be expressed by convergent series in integer or fractional powers of \(\epsilon\). The proof is based on the method of Newton diagrams. Under various additional conditions, various asymptotic formulas are obtained. The concrete cases of one or two Jordan chains are also investigated.