Discrete spectra of compactly perturbed bounded operators

Let C   be a bounded operator on a Banach space or on a Hilbert space, and consider the operator A=C+KA=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)−1zK(I−zC)−1(z∈C)(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.