A Schur or Schur-Agler function on the polydisc factors into two factors of the same class exactly when it has a realization whose colligation block matrix satisfies D21 = 0 and aD12 = C1B2, with the factors built explicitly from the matrix blocks.
Agler, On the representation of certain holomorphic functions defi ned on a polydisc , Topics in operator theory: Ernst D
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Factorizations of Schur functions
A Schur or Schur-Agler function on the polydisc factors into two factors of the same class exactly when it has a realization whose colligation block matrix satisfies D21 = 0 and aD12 = C1B2, with the factors built explicitly from the matrix blocks.