On the similarity of analytic matrix functions (Q2457875)

Taking \( L(\mathbb{C}^n)\)-valued analytic functions \(F\) and \(G\) which are similar on an open set from \(\mathbb{C}^n\), i.e., \( G(\omega) = X(\omega)F(\omega)X^{-1}(\omega)\), analyticity conditions for the invertible matrix \(X(\omega)\) are obtained. Generally, \(X(\omega)\) cannot be chosen to depend even continuously on \(\omega\). In this paper, \(G(\omega) = X(\omega)F(\omega)X^{-1}(\omega)\) for a finite set of values \(\omega\), and the equality holds up to some given order on this finite set. The author presents relevant results on the topic and a different approach to some of them. Also, as a consequence of the main result, some known results can be deduced.