A theorem on comparison of spectra, and the spectral asymptotics for a keldysh pencil

Suppose that H is a normal operator, the pencil L₀ (λ) = I- λⁿHⁿ has a discrete and positive spectrum in the domain Ω(2θ, R) = (λ: larg λ < 2θ, λ > R). and S(λ) is an operator-valued function that is holomorphic in Ω(2θ. R.) and small in comparison to L₀(λ) (in a certain sense). A theorem is proved on comparison of the spectra of L(λ) = L₀(λ)- S(λ) and L₀(λ), i.e., on an estimate of the difference N(r) — N₀(r), where N(r) (N₀(r)) is the distribution function of the spectrum of L(λ) (L₀(λ)) in Ω(θ, p) (p ≥ R). This result implies generalizations of theorems of Keldysh on the asymptotic behavior of the spectrum of a polynomial operator pencil.