Existence of minimal nonsquare \(J\)-symmetric factorizations for self-adjoint rational matrix functions. (Q1426297)

Let \(\Phi(\lambda)\), \(\lambda\) an independent complex variable, be a rational matrix function having Hermitian values on the imaginary axis. If \(J\) is a signature matrix, i.e., \(J\) is simultaneously Hermitian and unitary, then a factorization \(\Phi(\lambda)= W(\lambda)JW(-\overline\lambda)^*\) is called a \(J\)-spectral factorization. Here \(W(\lambda)\) is a rational matrix function, possibly of rectangular size. The \(J\)-spectral factorization is said to be minimal if the McMillan degree of \(\Phi(\lambda)\) is twice that of \(W(\lambda)\). The main result of the present paper establishes the existence of a minimal \(J'\)-spectral factorization for any rational matrix function \(\Phi(\lambda)\) having Hermitian values on the imaginary axis, and in addition having constant signature and constant pole signature along the imaginary axis. The proof is based on an extension of a given \(\Phi(\lambda)\) as above to a matrix function that has the same McMillan degree as \(\Phi(\lambda)\) does and admits a \(J\)-spectral factorization with square size factors. Connections are made with generalized Bezoutian matrices and common invariant zeros. Open problems are formulated.