Factorization of \(J\)-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions
The article is devoted to the factorization problem of \(J\)-unitary matrix polynomials on the real line. Setting \(J =\begin{pmatrix} 0&-1\\1&0\end{pmatrix}\), a \(J\)-unitary \(2\times2\) matrix polynomial on the unit circle has a unique factorization into elementary \(J\)-unitary matrix polynomials. The approach is based on the kernel \[ K_U(z,w)=\frac{J-U(z)JU(w)^*}{z-w^*},\text{ where }z,w\in C, z\neq w^*, \] which is Hermitian and has a finite number \(k\) of negative squares. With \(U(z)\) there is a finite-dimensional reproducing kernel Pontryagin space \(K(U)\). The corresponding difference quotient operator \(R_0\) has a unique chain of invariant subspaces that gives a unique \(U(z)\). For \(U(z)=\begin{pmatrix} a(z)&b(z)\\c(z)&d(z)\end{pmatrix}\) is a \(2\times2\) matrix polynomial which is \(J\)-unitary on \(R\) and \(U(z)^*JU(z)=J\), \(z\in \mathbb R\). We report some of the main results. Theorem 3.1. Assume that \(U(z)\in U_J\). Then: (i) \(U(z)\) is a normalized elementary factor if and only if it is of the form \(U(z)=I_2+p(z)uu^*J\), where \(u\in C^2\) satisfies \(u^*Ju=0\) and \(p(z)\) is a real polynomial with \(p(0)=0\). In this case: if \(p(z)=t_kz^k+\dots+t_1z\) with \(t_k\neq0\), then \(k=\dim K(U)\) and the negative index \(k\) of the Pontryagin space \(K(U)\) is given by the expression \[ k=\begin{cases} [k/2],&t_k>0,\\[(k+1)/2],&t_k<0.\end{cases} \] (ii) \(U(z)\) admits a unique minimal factorization of the form \[ U(z)=U_1(z) \dots U_n(z)U(0) \] with normalized elementary factors \(U_j(z)\), \(j=1,\dots,n\) and the \(J\)-unitary constant \(U(0)\). The factorization of unitary matrix polynomials is done using the Schur algorithm. For that reason, the Schur transformation of generalized Nevanlinna functions was completed.
scientific article; zbMATH DE number 2081251
scientific article