The range of linear fractional maps on the unit ball

Richman, Alexander

doi:10.1090/S0002-9939-02-06596-6

The range of linear fractional maps on the unit ball
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by Alexander E. Richman PDF

Proc. Amer. Math. Soc. 131 (2003), 889-895 Request permission

Abstract:

In 1996, C. Cowen and B. MacCluer studied a class of maps on $\mathbb C^N$ that they called linear fractional maps. Using the tools of Kreĭn 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 Kreĭn 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.

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