On Realizations of Rational Matrix Functions of Several Complex Variables

Abstract

Holomorphic functions of one complex variable with a non-negative imaginary part, and the related classes of J-contractive matrix functions (contractive in a space with an indefinite metric, defined in a standard way by a hermitian matrix J such that J 2 = I) were actively studied by many mathematicians in the past years. A series of problems of mathematical analysis and its applications lead to the necessity of studying analogous classes of holomorphic functions of several complex variables. Holomorphic functions with a non-negative imaginary part in a tubular domain over a cone and in the polydisk were studied by V.S. Vladimirov (see [Vla2], [V1a5]–[V1a8]) and by Vladimirov and Drozhzhinov [29]. In bounded strictly star-shaped domains, and in particular in the classical symmetric domains, they were studied by L.A. Aizenberg and Sh.A. Dautov [1]. W. Rudin (see [Rud]) gives a “parametrical” representation of scalar rational inner functions in the polydisk \(\mathbb{D}^n \). The parameter is an arbitrary polynomial non-vanishing in \(\mathbb{D}^n \).