On a class of entire matrix function equations (Q2644062)

This paper concerns matrix function equations of the form \(X(\lambda)B(\lambda) + D(\lambda)Y(\lambda) = G(\lambda)\) for \(\lambda \in \mathbb{C}\). The coefficient functions \(B\) and \(D\) are \(n \times n\) matrix functions given by \(B(\lambda) = I_n + \int_{- \omega}^0e^{i\lambda t}b(t)dt\) and \(D(\lambda) = I_n + \int_{0}^\omega e^{i\lambda t}d(t)dt\). The right hand side is assumed to be known and of the form \(G(\lambda) = \int_{-\omega}^\omega e^{i\lambda t}g(t)dt.\) Theorem: In order that the above equation is solvable it is necessary and sufficient that for each common zero \(\lambda_0\) of \(\det(B(\lambda))\) and \(\det(D(\lambda))\) the following holds: if \(\phi\) is a root function of \(B(\lambda)\) at \(\lambda_0\) of order \(p\) and \(\psi\) is a root function of \(D(\lambda)^T\) at \(\lambda_0\) of order \(q\), then the function \(\psi(\lambda)^TG(\lambda)\psi(\lambda)\) has a zero at \(\lambda_0\) of order at least \(\min\{p,q\}\).