Application of linear fractional transformation in problems of localization of matrix spectra and roots of polynomials

The paper investigates the possibilities of using linear fractional transformations in a number of problems that can be reduced to spectral dichotomy. More specifically, for the dichotomy of the imaginary axis, estimates are given for areas containing eigenvalues, methods for determining the absence of a matrix spectrum on a ray and a segment are described. A method for dividing a polynomial into two factors whose roots lie in the right and left half-planes is described and substantiated.