On real and “symplectic” meromorphic plus-matrix-function and corresponding linear fractional transformation

The basic result is: if linear fractional transformation with meromorphic in the unit disk nondegenerate matrix of coefficients $A(z)$ maps the class of holomorphic contractive matrix function into itself so that real (symmetric) matrix functions are transformed into real (symmetric) matrix functions then there exists a mеromorphic scalar function $\rho(z)$ such that $\rho^{-1}(z) A(z)$ is $j$-expansive real (“symplectic” or “antisymplectic”) matrix function.