Abstract. We study spectral properties of nonselfadjoint rank one perturbations of compact selfadjoint operators. The problems under consideration include completeness of eigenvectors, relations between completeness of the perturbed operator and its adjoint, and the spectral synthesis problem. We obtain new criteria for completeness and spectral synthesis in this class as well as a series of counterexamples which show that the spectral structure of rank one perturbations is, in general, unexpectedly rich and complicated.A parallel spectral theory is developed for one-dimensional singular perturbations of unbounded selfadjoint operators. Our approach is based on a functional model for this class which translates the properties of operators to completeness problems for systems of reproducing kernels and their biorthogonals in some spaces of analytic (entire) functions. M.S.C.(2010):Primary: 34L10 47B32, 47A55; Secondary: 47A45.

This paper concerns pure subnormal operators with nite rank self-commutator, which w e call subnormal operators of nite type. We analyze Xia's theory of these operators 21]-23] and give its alternative exposition. Our exposition is based on the explicit use of a certain algebraic curve i n C 2 , which w e call the discriminant curve of a subnormal operator, and the approach of dual analytic similarity models of 26]. We g i v e a complete structure result for subnormal operators ofnite type, which corrects and strenghtens the formulation that Xia gave in 23]. Xia claimed that each subnormal operator of nite type is unitarily equivalent to the operator of multiplication by z on a weighted vector H 2 -space over a \quadrature Riemann surface" (with a nite rank perturbation of the norm). We explain how this formulation can be corrected and show that, conversely, every \quadrature Riemann surface" gives rise to a family of subnormal operators. We prove that this family is parametrized by the so-called characters. As a departing point of our study, w e formulate a kind of scattering scheme for normal operators, which includes Xia's model as a particular case. Introduction.This paper is devoted to an alternative exposition of some aspects of Xia's theory of subnormal operators from a di erent viewpoint. We make use of the results of 25] and the approach of 26] and give new 623 624 D. V. Yakubovich

Abstract. We find a system of two polynomial equations in two unknowns, whose solution allows us to give an explicit expression of the conformal representation of a simply connected three-sheeted compact Riemann surface onto the extended complex plane. This function appears in the description of the ratio asymptotic of multiple orthogonal polynomials with respect to so-called Nikishin systems of two measures.

Abstract. We study spectral properties of one-dimensional singular perturbations of an unbounded selfadjoint operator and give criteria for the possibility to remove the whole spectrum by a perturbation of this type. A counterpart of our results for the case of bounded operators provides a complete description of compact selfadjoint operators whose rank one perturbation is a Volterra operator.M.S.C. (2000): Primary: 47A55; Secondary: 47B25, 47B07.