Carathéodory theorem for matrix-valued functions and the maximal jump of spectral functions in extension problems (Q1813386)

Let \({\mathfrak M}\) be a family of matrix functions holomorphic in the half- plane \(\Pi=\{z\in\mathbb{C}\mid\text{Im}(z)>0\}\) (or in the disk \(D=\{z\in\mathbb{C}\mid| z|<1\}\) with nonnegative imaginary parts and representable in the form \(f_{\mathcal E}(z)=[a_{11}(z){\mathcal E}(z)+a_{12}(z)][a_{21}(z){\mathcal E}(z)+a_{22}(z)]^{-1}\) where \(A(z)=\| a_{ik}(z)\|\) is a meromorphic matrix function with \(\text{det} A(z)\equiv 0\) and \({\mathcal E}(z){\mathcal E}^*(z)\leq I\). The author proves that there exists the limit \(\lim 2 | z-\zeta_ 0|[a_{22}(z)a^*_{22}(z)-a_{21}(z)a^*_{21}(z)]^{-1}\) for \(z\zeta_ 0\in\partial\Pi\) (or \(\partial D)\) and that this limit is the maximal value of jumps of Riesz-Herglotz measures of functions in \({\mathfrak M}\). An equivalent result is proved for another representation of \({\mathfrak M}\).