On a generalization of Rouché's theorem for trace ideals with applications for resonances of Schrödinger operators (Q583572)

Let \(f(z)=1+F(z)\) be an operator-valued function meromorphic on a simply connected domain \(\Gamma\) and analytic on its boundary \(\partial \Gamma\) such that F(z) belongs to the Schatten-von Neumann class \(I_ p\) with \(p\geq 1\). The behaviour under perturbation of the expression \[ N_{\Gamma}(F)=(2\pi i)^{-1}\quad trace(\oint_{\partial \Gamma}f'(z)f(z)^{-1} dz) \] is investigated. Note that, in the analytic case, \(N_{\Gamma}(F)\) counts the number of eigenvalues (-1) of F(z) in \(\Gamma\).