A-regular–A-singular factorizations of generalized J-inner matrix functions - Derkach, V. A.; Sukhorukova, O. (Methods of Functional Analysis and Topology, 2017)
Let J be an m× m signature matrix, i.e., J=J^*=J^-1. An m× m mvf (matrix valued function) W(λ) 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. μT=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 [18] 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_κ^r(J) of right generalized J-inner mvf’s introduced in DD09. 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.