The range of linear fractional maps on the unit ball

In 1996, C. Cowen and B. MacCluer studied a class of maps on that they called linear fractional maps. Using the tools of Krein spaces, it can be shown that a linear fractional map is a self-map of the ball if and only if an associated matrix is a multiple of a Krein contraction. In this paper, we extend this result by specifying this multiple in terms of eigenvalues and eigenvectors of this matrix, creating an easily verified condition in almost all cases. In the remaining cases, the best possible results depending on fixed point and boundary behavior are given.