Discrete spectra of compactly perturbed bounded operators

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Abstract

Let C be a bounded operator on a Banach space or on a Hilbert space, and consider the operator 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 (z∈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.

Original languageEnglish
Pages (from-to)1-16
Number of pages16
JournalJournal of Mathematical Analysis and Applications
Volume447
Issue number1
DOIs
StatePublished - 1 Mar 2017

Keywords

  • Perturbations
  • Quasinormed ideals
  • Schatten–von Neumann operators

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