Effective construction of a class of positive operators in Hilbert space, which do not admit triangular factorization

A class of non-factorable positive operators is constructed. As a result, pure existence theorems in the well-known problems by Ringrose, Kadison and Singer are substituted by concrete examples

Discrete skew selfadjoint canonical systems and the isotropic Heisenberg magnet model

A discrete analog of a skew selfadjoint canonical (Zakharov-Shabat or AKNS) system with a pseudo-exponential potential is introduced. For the corresponding Weyl function the direct and inverse problem are solved explicitly in terms of three parameter matrices. As an application explicit solutions are obtained for the discrete integrable nonlinear equation corresponding to the isotropic Heisenberg magnet model. State space techniques from mathematical system theory play an important role in the proofs

Discrete canonical system and non-Abelian Toda lattice: Backlund-Darboux transformation and Weyl functions

A version of the iterated B\"acklund-Darboux transformation, where Darboux matrix takes a form of the transfer matrix function from the system theory, is constructed for the discrete canonical system and Non-Abelian Toda lattice. Results on the transformations of the Weyl functions, insertion of the eigenvalues, and construction of the bound states are obtained. A wide class of the explicit solutions is given. An application to the semi-infinite block Jacobi matrices is treated.Comment: Second version: Section on the explicit solutions and results on the bound states and insertion of eigenvalues are added, the presentation is slightly change