On the multiplicity of Darlington realizations of contractive matrix- valued functions (Q1097487)

The problem of multiplicity of Darlington realizations of contractive matrix-valued functions is studied in this article. \({\mathcal S}\) denotes the class of contractive matrix-valued functions, i.e. functions S(z) \((z=re^{it})\) holomorphic in the unit disk \(D=\{z;| x| <1\}\), for which \(\| S(z)\| \leq 1\). A Darlington realization of S(z)\(\in {\mathcal S}\), of order \(n\times m\), is the representation of S(z) as the linear functional transformation \(S(z)=[\alpha (z)\epsilon +\beta (z)][\gamma (z)\epsilon +\delta (z)]^{-1}\) (z\(\in D)\) over a constant matrix \(\epsilon\in {\mathcal S}\), with a matrix of coefficients. \(A(z)=\left( \begin{matrix} \alpha (z)\\ \gamma (z)\end{matrix} \begin{matrix} \beta (z)\\ \delta (z)\end{matrix} \right)\) which is j-inner, i.e. A(z) satisfies i) \(A^*(z)jA(z)-j>0\) (z\(\in D)\); ii) \(A^*(\xi)jA(\xi)-j=0\) a.e. \((\xi =e^{it})\), where \(J=\left( \begin{matrix} -In\\ 0\end{matrix} \begin{matrix} 0\\ In\end{matrix} \right)\) and the symbol * stands for hermitian conjugation. Using results from \textit{D. Z. Arov} [Izv. Akad. Nauk SSSR, Ser. Mat. 37, 1299-1331 (1973; Zbl 0316.30036)], three cases are separately considered for matrix-valued functions belonging to a subclass \({\mathcal S}\pi \in {\mathcal S}:\) a) S(z) is inner, i.e. \(In-S^*(\xi)S(\xi)=0\) a.e.; b) S(z) is not inner and \(\det [In-S^*(\xi)S(\xi)]\neq 0\) a.e.; c) S(z) is not inner and \(\det [In-S^*(1/\bar z)S(z)]=0\) (z\(\in D).\) A unique expression, valid for the three cases, for the matrix of coefficients A(z) of a Darlington realization of S(z)\(\in {\mathcal S}\pi\) is obtained. This result allows to examine the problem of constructing the set of all possible realizations of S(z) not only in the case b) solved in the paper cited above, but also in case c) proposed as an open problem.