Factorization formulas for some classes of generalized $J$-inner matrix valued functions
Abstract
The class $\mathcal{U}_\kappa(j_{pq})$ of generalized $j_{pq}$-inner matrix valued functions (mvf's) %and its subclass $\mathcal{U}^r_\kappa(j_{pq})$ was introduced in [2]. For a mvf $W$ from a subclass $\mathcal{U}^r_\kappa(j_{pq})$ of $\mathcal{U}_\kappa(j_{pq})$ the notion of the right associated pair was introduced in [13] and some factorization formulas were found. In the present paper we introduce a dual subclass $\mathcal{U}^\ell_\kappa(j_{pq})$ and for every mvf $W\in \mathcal{U}^\ell_\kappa(j_{pq})$ a left associated pair $\{\beta_1,\beta_2\}$ is defined and factorization formulas for $W$ in terms of $\beta_1,\beta_2$ are found. The notion of a singular generalized $j_{pq}$-inner mvf $W$ is introduced and a characterization of singularity of $W$ is given in terms of associated pair.
Key words: J-inner matrix valued function, generalized Schur class, Kre˘ın-Langer factorization, Potapov-Ginzburg transform, reproducing kernel space, associated pair.
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