On the distribution of the discrete spectrum of nuclearly perturbed operators in Banach spaces

Abstract

Let Z 0 be a bounded operator in a Banach space X with purely essential spectrum and K a nuclear operator in X. Using methods of complex analysis we study the discrete spectrum of Z 0 + K and derive a Lieb-Thirring type inequality. We obtain estimates for the number of eigenvalues in certain regions of the complex plane and an estimate for the asymptotics of the eigenvalues approaching to the essential spectrum of Z 0 .Mathematics Subject Classifictaion (2010): 47A75, 47A10, 47A55, 47B10.

Perturbation determinants in Banach spaces — with an application to eigenvalue estimates for perturbed operators

“…Of course, to obtain a bound of the above form we would need much more information on the operators involved. We will not further pursue this method in the present paper and simply refer to [4] for some results in this direction (for the Hilbert space setting, see also [6]).…”

“…The theory of perturbation determinants and regularized perturbation determinants for Hilbert space operators is very well-developed and has been successfully applied in a large variety of settings (see, e.g., [14], [32], [33], [35], [9], [6] and references therein). On the other hand, with two recent exceptions [4,5], regularized perturbation determinants for operators on general Banach spaces seem to have been hardly considered at all. It is the aim of this paper to provide some first results for a corresponding theory.…”

The Generalized Birman-Schwinger Principle

“…While most of these sources focus on self-adjoint situations, eventually, the Birman-Schwinger technique was extended to non-self-adjoint situations in the context of complex resonances in [1,74], and systematically in [30] (and later again in [5], [6], [29]), adapting factorization techniques developed in Kato [46], Konno and Kuroda [51], and Howland [45], primarily in the self-adjoint context. The past 15 years saw enormous interest in various aspects of spectral theory associated with non-self-adjoint problems and we refer, for instance, to [10,12,14,15,16,19,21,22,23,24,25,26,44,56,67,68], again, just a tiny selection of the existing literature without hope of any kind of completeness, and to [9, Sect. III.9], [54] for applications to spectral stability of nonlinear systems.…”