The Schur class, denoted by \({\mathcal {S}}({\mathbb {D}})\), is the set of all functions analytic and bounded by one in modulus in the open unit disc \({\mathbb {D}}\) in the complex plane \({\mathbb {C}}\), that is

The elements of \({\mathcal {S}}({\mathbb {D}})\) are called Schur functions. A classical result going back to I. Schur states: A function \(\varphi : {\mathbb {D}} \rightarrow {\mathbb {C}}\) is in \({\mathcal {S}}({\mathbb {D}})\) if and only if there exist a Hilbert space \({\mathcal {H}}\) and an isometry (known as colligation operator matrix or scattering operator matrix)

such that \(\varphi \) admits a transfer function realization corresponding to V, that is

An analogous statement holds true for Schur functions on the bidisc. On the other hand, Schur-Agler class functions on the unit polydisc in \({\mathbb {C}}^n\) is a well-known “analogue” of Schur functions on \({\mathbb {D}}\). In this paper, we present algorithms to factorize Schur functions and Schur-Agler class functions in terms of colligation matrices. More precisely, we isolate checkable conditions on colligation matrices that ensure the existence of Schur (Schur-Agler class) factors of a Schur (Schur-Agler class) function and vice versa.

Schur类用\（{\ mathcal {S}}（{\ mathbb {D}}）\）表示，是所有函数的集合，在开放单位圆盘\（{\ mathbb {D}} \）在复平面\（{\ mathbb {C}} \\）中，即

的元素\（{\ mathcal {S}}（{\ mathbb {d}}）\）被称为舒尔功能。一个经典的结果可以回溯到I. Schur指出：函数\（\ varphi：{\ mathbb {D}} \ rightarrow {\ mathbb {C}} \}位于\（{\ mathcal {S}}（{\ mathbb {D}}）\）并且仅当存在希尔伯特空间\（{\ mathcal {H}} \）和等轴测图（称为定理算子矩阵或散射算子矩阵）时

这样\（\ varphi \）允许对应于V的传递函数实现，即

一个类似的陈述适用于bidisc上的Schur函数。另一方面，\（{\ mathbb {C}} ^ n \）中单元多圆盘上的Schur-Agler类函数是\（{\ mathbb {D}} \\ ）。在本文中，我们提出了根据合积矩阵分解Schur函数和Schur-Agler类函数的算法。更准确地说，我们在合数矩阵上隔离可检查的条件，以确保存在Schur（Schur-Agler类）函数的Schur（Schur-Agler类）因子，反之亦然。