Generalized solutions of Riccati equalities and inequalities

The Riccati inequality and equality are studied for infinite dimensional linear discrete time stationary systems with respect to the scattering supply rate. The results obtained are an addition to and based on our earlier work on the Kalman-Yakubovich-Popov inequality in [6]. The main theorems are closely related to the results of Yu. M. Arlinski\u{\i} in [3]. The main difference is that we do not assume the original system to be a passive scattering system, and we allow the solutions of the Riccati inequality and equality to satisfy weaker conditions.Comment: Published in Methods of Functional Analysis and Topology (MFAT), available at http://mfat.imath.kiev.ua/article/?id=84

Szegö-Kac-Achiezer formulas in terms of realizations of the symbol

AbstractFor rational and analytic matrix functions new formulas are obtained for the limits in the Szegö-Kac-Achiezer limit theorems. In the rational case the new expressions are given in terms of finite matrices which come from special representations of the matrix functions. These representations are known as realizations in mathematical systems theory

A time-varying generalization of the canonical factorization theorem for Toeplitz operators

AbstractThe canonical factorization theorem for the symbol of a Toeplitz operator is generalized to a class of non-Toeplitz operators. The operators in this class may be described as input-output operators of time-varying linear systems. Dichotomy of difference equations plays an important role

Column reduced rational matrix functions with given null-pole data in the complex plane

AbstractExplicit formulas are given for rational matrix functions which have a prescribed null-pole structure in the complex plane and are column reduced at infinity. A full parametrization of such functions is obtained. The results are specified and developed further for matrix polynomials

Minimal representations of semiseparable kernels and systems with separable boundary conditions

AbstractThe simplest representations of triangular parts of finite rank kernels are analysed. Criteria for uniqueness up to similarity are given. The results are applied to the problem of minimal realization of systems with separable boundary conditions

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

Semiseparable integral operators and explicit solution of an inverse problem for the skew-self-adjoint Dirac-type system

Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form $\phi(\lambda)\exp\{-2i\lambda D\}$, where $\phi$ is a strictly proper rational matrix function and $D=D^* \geq 0$ is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case