Factorizations of Schur functions

Article Type

Research Article

Publication Title

Complex Analysis and Operator Theory


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)={φ∈H∞(D):‖φ‖∞:=supz∈D|φ(z)|≤1}.The elements of S(D) are called Schur functions. A classical result going back to I. Schur states: A function φ: 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=[aBCD]:C⊕H→C⊕H,such that φ admits a transfer function realization corresponding to V, that is φ(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.



Publication Date



Open Access, Green

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