Discrete spectra of compactly perturbed bounded operators

详细信息    查看全文

• 作者：Michael Gil' ^{gilmi@bezeqint.net" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Schatten&ndash ; von Neumann operators ; Perturbations ; Quasinormed ideals
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：447
• 期：1
• 页码：1-16
• 全文大小：367 K

文摘

Let C   be a bounded operator on a Banach space or on a Hilbert space, and consider the operator A=C+K$A=C+K$, where K is a compact operator. We are interested in the discrete spectrum of A in domains free of the spectrum of C. In the first part of the paper we deal with Hilbert space operators assuming that K is a Schatten–von Neumann operator. Besides, the bounds for the absolute values and imaginary parts of the eigenvalues of A are obtained in terms of the Schatten–von Neumann norm of K and norm of the resolvent of C. In addition, we estimate the counting functions of the numbers of the eigenvalues of A in various domains. In the second part we particularly generalize our results to so called p  -quasi-normed ideals of compact operators in a Banach space. Our main tool is a combined usage of the regularized determinant of the operator zK(I−zC)⁻¹$zK{(I-zC)}^{-1}$(z∈C)$(z\in \mathbb{C})$, where I is the unit operator, and recent norm estimates for resolvents. Applications of our results to the non-selfadjoint Jacobi operator are also discussed.