Remarks on the perturbation of analytic matrix functions. II (Q1824178)

Let A(\(\lambda)\), B(\(\lambda\),\(\epsilon)\) be \(n\times n\) matrix functions, holomorphic near \(\lambda =\lambda_ 0\) and \(\epsilon =0\), \(B(\lambda,0)=0\), and let \(\lambda_ 0\) be an eigenvalue of A(\(\lambda)\) with partial multiplicities \(m_ 1,...,m_ g\). Using the Newton diagram, a sufficient condition is given which ensures that for each partial multiplicity \(m_ j\) there are \(m_ j\) eigenvalues \(\lambda_{j\sigma}(\epsilon)\) of \(T(\lambda,\epsilon)=A(\lambda)+B(\lambda,\epsilon)\) with an expansion \(\lambda_{j\sigma}(\epsilon)=\lambda_ 0+\alpha_{j\sigma}\epsilon^{1/m_ j}+o(| \epsilon |^{1/m_ j})\) for small \(\epsilon\), \(\alpha_{j\sigma}\neq 0\); \(\sigma =1,...,m_ j\), \(j=1,...,g\). This result is specialized to hermitian matrix functions and \(\lambda\)-linear problems. This note is a correction and extension of the earlier work by the second author: Integral Equations Oper. Theory 9, 592-599 (1986; Zbl 0605.47013).