A~theorem on comparison of spectra, and spectral asymptotics for a~Keldysh pencil

A. S. Markus ; V. I. Matsaev

Sbornik. Mathematics, Tome 51 (1985) no. 2, pp. 389-404

Résumé

Suppose that $H$ is a normal operator, the pencil $L_0(\lambda)=I-\lambda^nH^n$ has a discrete and positive spectrum in the domain $\Omega(2\theta,R)=\{\lambda:\lvert\arg\lambda\rvert2\theta,\ |\lambda|>R\}$, and $S(\lambda)$ is an operator-valued function that is holomorphic in $\Omega(2\theta,R)$ and small in comparison to $L_0(\lambda)$ (in a certain sense). A theorem is proved on comparison of the spectra of $L(\lambda)=L_0(\lambda)-S(\lambda)$ and $L_0(\lambda)$, i.e., on an estimate of the difference $N(r)-N_0(r)$, where $N(r)$ ($N_0(r)$) is the distribution function of the spectrum of $L(\lambda)$ ($L_0(\lambda)$) in $\Omega(\theta,\rho)$ ($\rho\geqslant R$). This result implies generalizations of theorems of Keldysh on the asymptotic behavior of the spectrum of a polynomial operator pencil. Bibliography: 14 titles.