The failure of factorization of positive matrix functions on noncircular contours (Q1336410)

Every simple closed contour divides the extended plane into two domains: a bounded domain \(G_ +\) and an unbounded domain \(G_ -\). Denote by \(C (\Gamma)\) the set of all continuous functions on \(\Gamma\), let \(C^ + (\Gamma)\) and \(C^ - (\Gamma)\) be the subsets of \(C(\Gamma)\) consisting of the functions which admit analytic extensions to \(G_ +\) and \(G_ -\) respectively. For \(n \times n\)-matrix functions and \(n\)-vector functions the notations \((C^ \pm (\Gamma))_{n \times n}\) and \((C^ \pm (\Gamma))_ n\) are used. An \(n \times n\)-matrix function \(A(t)\) continuous on \(\Gamma\) is said to admit a factorization with respect to \(\Gamma\) if there exists a matrix function \(A_ \pm (t) \in (C^ \pm (\Gamma))_{n \times n}\) such that \(\text{det} A_ + (t) \neq 0\) for all \(t \in G_ + \cup \Gamma\), \(\text{det} A_ - (t) \neq 0\) for all \(t \in G_ - \cup \Gamma\), \(A_ - (\infty) = I\), and \(A(t) = A_ + (t)A_ - (t)\) \((t \in \Gamma)\). It is well known that any Hölder continuous matrix function positive on a circle admits the factorization. In this paper, the authors show that this property is characteristic for the circle: If every matrix function of second order that is positive and Hölder continuous on \(\Gamma\) admits the factorization, then \(\Gamma\) is a circle.