Sukhorukova O. Generalized $J$-inner and $\gamma$-generating matrix valued functions and indefinite interpolation problems

The thesis is devoted to the study of generalized $J$-inner matrix valued functions and generalized $\gamma$-generating matrices and to solving of the Takagi-Sarason and Nehari-Takagi problems. New classes of left generalized $j_{pq}$-inner matrix valued functions are introduced and their connection with the classes of right generalized $j_{pq}$-inner matrix valued functions is studied. The notion of singular and regular generalized $j_{pq}$-inner matrix valued functions are introduced and characterization of singular matrix valued functions is found. Sufficient conditions for the regularity of a generalized $j_{pq}$-inner matrix valued function and a criterion for the existence of regular-singular factorization for a rational generalized $j_{pq}$-inner matrix valued function are found. An example of a right generalized $j_{pq}$-inner matrix valued function $W$ which does not admit a regular-singular factorization is given. Another class of matrix valued functions that is studied in the thesis is the class of generalized $\gamma$-generating matrices. We introduce definitions of right and left generalized $\gamma$-generating matrices. A connection between generalized $\gamma$-generating matrices and generalized $j_{pq}$-inner matrix valued functions is established. A criterion of existence of regular-singular factorization for a rational generalized $\gamma$-generating matrices is found. The Takagi-Sarason problem and the Nehari-Takagi problem are studied in the thesis. A full description of solutions of the Takagi-Sarason interpolation problem is found. Under certain mild assumption, a one-to-one correspondence between solutions of the Nehari-Takagi problem and solutions of some Takagi-Sarason interpolation problem is established. A description of solutions of completely indeterminate Nehari-Takagi problem for bounded functions which allow pseudocontinuations is obtained.