We investigate the analytic perturbation of generalized inverses. Firstly we analyze the analytic perturbation of the Drazin generalized inverse (also known as reduced resolvent in operator theory). Our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent. It allows us to treat regular and singular perturbations in a unified framework. W...

We study perturbations of the Drazin inverse of a closed linear operator A for the case when the perturbed operator has the same spectral projection as A. This theory subsumes results recently obtained by Wei and Wang, Rakočević and Wei, and Castro and Koliha. We give explicit error estimates for the perturbation of Drazin inverse, and error estimates involving higher powers of the operators.

The two-dimensional Hamiltonian system (∗) y′(x) = zJH(x)y(x), x ∈ (a,b), where the Hamiltonian H takes non-negative 2× 2-matrices as values, and J := ( 0 −1 1 0 ) , has attracted a lot of interest over the past decades. Special emphasis has been put on operator models and direct and inverse spectral theorems. Weyl theory plays a prominent role in the spectral theory of the equation, relating t...

In this paper we study the W -weighted Drazin inverse of the bounded linear operators between Banach spaces and its representation theorem. Based on this representation, utilizing the spectral theory of Banach space operators, we derive an approximating expression of the W -weighted Drazin inverse and an error bound. Also, a perturbation theorem for the W -weighted Drazin inverse is uniformly o...

Inverse scattering and spectral one-dimensional problems are discussed systematically in a self-contained way. Many novel results due to the author are presented. The classical results are often presented in a new way. Several highlights of the new results include: 1) Analysis of the invertibility of the steps in the Gel’fand-Levitan and Marchenko inversion procedures, 2) Theory of the inverse ...

We provide a rigorous treatment of the inverse scattering transform for the entire Toda hierarchy. In addition, we revisit the connection between trace formulas and conserved quantities from the viewpoint of Krein’s spectral shift theory.