Factorizations of Schur functions

The Schur class, denoted by S(D), is the set of all functions analytic and bounded by one in modulus in the open unit disc D in the complex plane C, that is S( D) = similar to.. H 8 (D) : similar to. similar to 8 := sup z. D |.(z)| = 1 similar to. The elements of S( D) are called Schur functions. A classical result going back to I. Schur states: A function. : D. C is in S( D) if and only if there exist a Hilbert space H and an isometry (known as colligation operator matrix or scattering operator matrix) V = similar to a B C D similar to : C. H. C. H, such that. admits a transfer function realization corresponding to V, that is.(z) = a + zB( IH - zD)-1C (z. D). An analogous statement holds true for Schur functions on the bidisc. On the other hand, Schur-Agler class functions on the unit polydisc in Cn is a well-known "analogue" of Schur functions on 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.