An analogue of Rouché's theorem in the generalized Smirnov class (Q2901659)

The possibility of extending Rouché's theorem in matrix-operator sense is considered in the present article. As is well-known, holomorphic functions \(F\) and \(G\) defined in the closed unit disk \(\overline{{\mathbb D}}\) in \({\mathbb C}\) that satisfy the inequality \(| F| >| G| \) on the unit circle have the same number of zeros in \({\mathbb D}\). Here, the author extends the above theorem to matrix-valued holomorphic functions:NEWLINENEWLINEDenote by \({\mathcal M}(F)\) the summarized multiplicity of the meromorphic matrix function \(F\) in \({\mathbb D}\). Suppose that \(\det(F+G)\not\equiv 0\) in \({\mathbb D}\) and \(\| GF^{-1}\|\leq 1\) on \(\partial {\mathbb D}\). Then \({\mathcal M} (F+G)\leq {\mathcal M} (F)\). If, in addition, the restriction of \(F(F+G)^{-1}\) to \(\partial {\mathbb D}\) is integrable on \(\partial {\mathbb D}\), then the equality \({\mathcal M} (F+G) = {\mathcal M} (F)\) holds.