\(J\)-unitary preserving automorphisms of rational matrix functions: State space theory, interpolation, and factorization
Let \(GL_ m({\mathcal R})\) be the multiplicative group of \(C^{m\times m}\) valued rational functions of one complex variable \(z\) with determinant not vanishing identically. According to state space representations, the author analyzes general inner automorphisms of \(GL_ m({\mathcal R})\) that preserve the set of matrix functions which are unitary (with respect to an indefinite scalar product) for real, or unimodular values of \(z\). As examples, the author presents a number of generalized inner automorphisms where \(GL(J,\Delta)\) with \(\Delta=R\) or \(\Delta=T\) for various choices of \(J\) is left invariant. Moreover, the author studies the homogeneous interpolation problem for rational matrix functions belonging to the associated \(J\)-unitary group with a given automorphism \({\mathcal A}\) of one of the four forms. In the end, as a development, the author characterizes the factorizations of rational matrix functions \(W\) in the associated \(J\)-unitary group such that both \(W=W_ 1W_ 2\) and \({\mathcal A}(W)={\mathcal A}(W_ 1){\mathcal A}(W_ 2)\) are minimal \(J\)-unitary factorizations.
scientific article; zbMATH DE number 553975