On perturbation theory for points of discrete spectrum of dissipative operator functions

We consider the operator function L(α, θ) = A(α) − θR of two complex arguments, where A(α) is an analytic operator function defined in some neighborhood of a real point α0 ∈ ℝ and ranging in the space of bounded operators in a Hilbert space ℋ. We assume that A(α) is a dissipative operator for real α in a small neighborhood of the point α0 and R is a bounded positive operator; moreover, the point α0 is a normal eigenvalue of the operator function L(α, θ0) for some θ0 ∈ ℝ, and the number θ0 is a normal eigenvalue of the operator function L(α0θ). We obtain analogs and generalizations of well-known results for self-adjoint operator functions A(α) on the decomposition of α- and θ-eigenvalues in neighborhoods of the points α0 and θ0, respectively, and on the representation of the corresponding eigenfunctions by series.