Matricial Nehari Problems, J-Inner Matrix Functions and the Muckenhoupt Condition

The classes of regular and strongly regular γ-generating matrices and J-inner matrix valued functions arose in the investigation of the matricial Nehari problem, bitangential interpolation problems, and inverse problems for canonical systems as well as the theory of characteristic functions of operators and operator nodes. In this paper, new characterizations of these classes are developed. In particular, the property of strong regularity is characterized in terms of a matricial Muckenhoupt (A2) condition in the Treil–Volberg form. These results are based on parametrizations that are intimately connected with Darlington representations of matrix valued functions in the Schur and Carathéodory classes. As a byproduct of this analysis, examples of strongly regular γ-generating matrices and entire J-inner matrix valued functions that are unbounded on the circle and the real line, respectively, are presented.