Nonself-adjoint operators with almost Hermitian spectrum: Matrix model. I

Nonself-adjoint, nondissipative perturbations of bounded self-adjoint operators with real purely singular spectrum are considered. Using a functional model of a nonself-adjoint operator as a principal tool, spectral properties of such operators are investigated. In particular, in the case of rank two perturbations the pure point spectral component is completely characterized in terms of matrix elements of the operators’ characteristic function.