$A$-regular–$A$-singular factorizations of generalized $J$-inner matrix functions
Abstract
Let $J$ be an $m\times m$ signature matrix, i.e., $J=J^*=J^{-1}$. An $m\times m$ mvf (matrix valued function) $W(\lambda)$ that is meromorphic in the unit disk $\mathbb{D}$ is called $J$-inner if $W(\lambda)JW(\lambda)^*\leq J$ for every $\lambda$ from $\mathfrak{h}_W^+$, the domain of holomorphy of $W$, in ${\mathbb{D}}$, and $W(\mu)JW(\mu)^*= J$ for a.e. $\mu\in\mathbb{T}=\partial \mathbb{D}$. A $J$-inner mvf $W(\lambda)$ 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 ${\mathcal U}_\kappa^r(J)$ of right generalized $J$-inner mvf's introduced in~\cite{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.
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