Fredholm analytic operator families and perturbation of resonances. (English) Zbl 1113.47010

The author considers an operator-valued function \(K (\lambda,\alpha)\) which is analytic in both variables and takes values in the class of compact operators defined on a Hilbert space \(\mathcal{H}\). For a fixed parameter \(\alpha_{0}\), the point \(\lambda_{0}\) is called singular for the operator-valued function \(K (\lambda,\alpha_{0})\) if \(\ker (K (\lambda_{0},\alpha_{0}) - I) \neq 0.\) Denote \(K (\lambda) = K (\lambda,\alpha_{0})\), and suppose that \(\dim \ker (K (\lambda_{0}) - I) = 1\) and the resolvent \((K (\lambda) - I)^{-1}\) has a pole of order \(m\) at \(\lambda_{0}\).
The author proves that if \((K_{\lambda}^{'} (\alpha_{0},\alpha_{0}) \varphi_{0},\psi_{0}) \neq 0\) for eigenvectors \(\varphi_{0} \in \ker (K (\lambda_{0},\alpha_{0}) - I)\) and \(\psi_{0} \in \ker (K^{*} (\lambda_{0},\alpha_{0}) - I)\), then for \(\alpha\) close enough to \(\alpha_{0}\), the singular points of \(K(\lambda,\alpha)\) from a sufficiently small neighborhood of \(\lambda_{0}\) are represented by the Puiseux power expansions in the variable \((\alpha - \alpha_{0})^{1/m}\). All these singular points are necessarily simple and can be enumerated in such a way that \[ \lambda_{s} (\alpha) = \lambda_{0} + \lambda_{s}^{(1)} (\alpha - \alpha_{0})^{1/m} + \dots, \quad s = 0,1,\dots,m - 1, \] where \(\lambda_{s}^{(1)} = (- (K_{\alpha}^{'} (\lambda_{0},\alpha_{0}) \varphi_{0},\psi_{0}))^{1/m} e^{2i \pi s/m}\).
Also, an analogous result is proved for the case in which \(\dim (K (\lambda_{0}) - I) = n\) and \((K (\lambda) - I)^{-1}\) has a simple pole at the point \(\lambda_{0}\).


47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47A55 Perturbation theory of linear operators
58B15 Fredholm structures on infinite-dimensional manifolds
47A10 Spectrum, resolvent
47A53 (Semi-) Fredholm operators; index theories