A-REGULAR-A-SINGULAR FACTORIZATIONS OF GENERALIZED J-INNER MATRIX FUNCTIONS

摘要

Let J be an m × m signature matrix, i.e., J = J~* = J~(-1). An m × m mvf (matrix valued function) W that is meromorphic in the unit disk D is called J-inner if W(λ)JW(λ)~*≤ J for every λ from h_W~+, the domain of holomorphy of W, in D, and W(μ)JW(μ)~* = J for a.e. μ ? T = ?D. A J-inner mvf W(λ) is called A-singular if it is outer and it is called right A-regular if it has no non-constant A- singular right divisors. As was shown by D. Arov [8] every J-inner mvf admits an essentially unique A-regular-A-singular factorization W = W~((1))W~((2)). In the present paper this factorization result is extended to the class U_k~r(J) of right generalized J- inner mvf's introduced in [18]. The notion and criterion of A-regularity for right generalized J-inner mvf's are presented. The main result of the paper is that we find a criterion for existence of an A-regular-A-singular factorization for a rational generalized J-inner mvf.