Rational matrix functions with coisometric values on the imaginary line (Q1900421)

Given two signature matrices \(J_1\) and \(J_2\), the authors provide a necessary and sufficient condition for a rational matrix \(W\), and holomorphic at infinity, to satisfy the equation \(J_1= W(z)J_2 W(z)^*\) on the imaginary axis. Here a signature matrix is a matrix \(J\) that satisfies \(J= J^*= J^{-1}\). The present condition is based on a Lyapunov equation involving matrices \((A,B,C,D)\) in an observable realization \((A,B,C,D)\) of \(W\), and it extends a more known condition when \(J_1= J_2= J\), in which case \(W(z)\) is \(J\)-unitary on the imaginary line. The fact that \(J_1\neq J_2\) is possible means, in particular, that \(J_1\) and \(J_2\) may have different sizes, and hence \(W\) is allowed to be rectangular. When the condition is satisfied, then for each observable realization \((A,B,C,D)\) of \(W\) there exists a unique, possibly singular, Hermitian matrix \(G\) which satisfies the Lyapunov equation and such that \(CG= -DJ_2B^*\). Finally, the authors also indicate that the imaginary axis, where the equation is satisfied, can be replaced by the unit circle or a more general curve as considered in \textit{D. Alpay} and \textit{H. Dym} [J. Funct. Anal. 111, No. 1, 1-28 (1993; Zbl 0813.46018)].