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**On linear fractional transformations associated with generalized \(J\)-inner matrix functions.**
*(English)*
Zbl 1185.47015

The present paper is a wonderful work dedicated to generalized Schur classes and the theory of indefinite inner product spaces. Comprising 50 pages, the paper is organized in four big sections. The first one is a necessary introduction leading the reader into the topic and presenting the main results of the paper. In the second section, the basic notions of left and right coprime factorizations are introduced, and the connection with the Kreĭn-Langer factorizations of generalized Schur functions is discussed. Also, an extension of the theory of reproducing kernel Pontryagin spaces associated with a generalized Schur function is presented.

The first main result of the paper (Theorem 1.1), presented in the introduction, is proved in the third section. This result characterizes the range of a linear fractional transformation \(T_W\) associated with a function \(W\) which belongs to the class of generalized \(J\)-inner matrix valued functions. Factorization formulas for matrix valued functions are obtained in the fourth section of the paper, which will help in characterization of some reproducing kernel Pontryagin spaces. Also, using the obtained results from the fourth section and based on the main Theorem 1.1 as well as a special case of Kreĭn-Langer generalization of Rouché’s Theorem, the proof of the parametrization given by the second main result (Theorem 1.2) is given here.

The first main result of the paper (Theorem 1.1), presented in the introduction, is proved in the third section. This result characterizes the range of a linear fractional transformation \(T_W\) associated with a function \(W\) which belongs to the class of generalized \(J\)-inner matrix valued functions. Factorization formulas for matrix valued functions are obtained in the fourth section of the paper, which will help in characterization of some reproducing kernel Pontryagin spaces. Also, using the obtained results from the fourth section and based on the main Theorem 1.1 as well as a special case of Kreĭn-Langer generalization of Rouché’s Theorem, the proof of the parametrization given by the second main result (Theorem 1.2) is given here.

Reviewer: Ilie Valuşescu (Bucureşti)

### MSC:

47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |

46C20 | Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.) |

46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |

47A57 | Linear operator methods in interpolation, moment and extension problems |

47B20 | Subnormal operators, hyponormal operators, etc. |