TY - JOUR

T1 - Discrete spectra of compactly perturbed bounded operators

AU - Gil, Michael

N1 - Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2017/3/1

Y1 - 2017/3/1

N2 - 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.

AB - 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.

KW - Perturbations

KW - Quasinormed ideals

KW - Schatten–von Neumann operators

UR - http://www.scopus.com/inward/record.url?scp=84994155146&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2016.10.005

DO - 10.1016/j.jmaa.2016.10.005

M3 - Article

AN - SCOPUS:84994155146

VL - 447

SP - 1

EP - 16

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -