Generalized J-inner and γ-generating matrix-valued functions and indefinite interpolation problems

In the 50s M. S. Livšic introduced the concept of a characteristic function of a non-unitary linear operator which plays a crucial role in the spectral theory of such operators. An important property of the characteristic function is that it is J-contractive in the unit disc for some signature matrix J. The physical analog of the characteristic function in the theory of linear systems is the transfer function of such system. As was shown by B. Francis and K. Glover, the reduction problem in the theory of H-control is equivalent to a Nehari-Takagi problem. The resolvent matrix of the Nehari-Takagi problem belongs to the class of the so-called generalized γ-generating matrices which will be one of the main objects of this talk. In the present talk the general-matrix Nehari-Takagi problem will be considered, and under some extra assumptions it will be shown that this problem can be reduced to the Takagi-Sarason interpolation problem studied earlier by V. Derkach and H. Dym. This connection allows a parameterization of the solutions of the Schur-Takagi problem. The subclasses of regular and singular generalized J-inner matrix-valued functions will be considered as well.