Abstract. In this paper, we describe a class of maps of the unit ball in ${\msbm C}^N$ into itself that generalize the automorphisms and deserve to be called linear fractional maps. They are special cases or generalizations of the linear fractional maps studied by Kre?n and Smul'jan, Harris and others. As in the complex plane, a linear fractional map on ${\msbm C}^N$ is represented by an $(N+1)\times(N+1)$ matrix. Basic connections between the properties of the map and the properties of this matrix viewed as a linear transformation on an associated Kre?n space are established. These maps are shown to induce bounded composition operators on the Hardy spaces $H^p({\msbm B}_N)$ and some weighted Bergman spaces and we compute the adjoints of these composition operators on these spaces. Finally, we solve Schroeder's equation $f\circ\varphi = \varphi '(0)f$ when $\varphi $ is a linear fractional self-map of the ball fixing 0.

AMS Subject Classification (1991): 47B38, 32A30, 30C45

Received September 2, 1998, and in revised form March 22, 1999. (Registered under 2739/2009.)