Translation of eigenvalues of a dissipative bundle of operators (Q1326088)

Let \(L(\alpha, \theta)= \alpha^ n A_ n+ \alpha^{n- 1} A_{n- 1}+\cdots+ A_ 0- \theta I\) be a two-parameter bundle of operators in a Hilbert space \(H\). Fix a pair of real numbers \((\alpha_ 0, \theta_ 0)\) such that \(\dim\ker L(\alpha_ 0, \theta_ 0)\geq 1\), and either \(\theta_ 0\) is a normal eigenvalue of the operator function \(L(\alpha_ 0, \theta)\) or \(\alpha_ 0\) is a normal eigenvalue of \(L(\alpha, \theta_ 0)\). This paper concerns the behaviour of the eigenvalues and corresponding eigenvectors as functions of the two parameters assuming that the imaginary part of \(L(\alpha, \theta_ 0)\) is positive semi-definite for each real \(\alpha\) in some open ball around \(\alpha_ 0\). Two theorems are given without proof. The first theorem is a Rellich type theorem, and concerns the case when \(\theta\) is considered to be the spectral parameter and \(\alpha\) the perturbation argument. In the second theorem \(\alpha\) is the spectral parameter and \(\theta\) the perturbation argument.