Factorizations of Schur functions

The Schur class, denoted by $${\mathcal {S}}({\mathbb {D}})$$ , is the set of all functions analytic and bounded one in modulus open unit disc $${\mathbb {D}}$$ complex plane {C}}$$ that $$\begin{aligned} {\mathcal {D}}) = \left\{ \varphi \in H^\infty ({\mathbb {D}}): \Vert _{\infty } := \sup _{z {\mathbb {D}}} |\varphi (z)| \le 1\right\} . \end{aligned}$$ elements are called functions. A classical result going back to I. states: function $$\varphi : {D}} \rightarrow if only there exist a Hilbert space {H}}$$ an isometry (known as colligation operator matrix or scattering matrix) V \begin{bmatrix} &{}\quad B \\ C D \end{bmatrix} {C}} \oplus {H}} {H}}, such $$ admits transfer realization corresponding V, (z) + z (I_{{\mathcal {H}}} - D)^{-1} \quad (z {D}}). An analogous statement holds true for on bidisc. On other hand, Schur-Agler class polydisc {C}}^n$$ well-known “analogue” In this paper, we present algorithms factorize terms matrices. More precisely, isolate checkable conditions matrices ensure existence (Schur-Agler class) factors vice versa.