A class of linear fractional maps of the ball and their composition operators (Q868917)

Let \(A = (a_{i,j})\) be an \(N \times N\)-matrix, \(B + (b_j), \mathbb C = (c_j)\) be \(N\)-column vectors, and \(d\) be a complex number. A linear fractional map of \(\mathbb C^n\) is a map of the form \[ u(z) = \frac{Az + B}{\langle z,\overline{\mathbb C}\rangle + d}, \] where \(\langle.,.\rangle\) is the Hermitian product in \(\mathbb C^n.\) The author investigates properties of linear fractional self-maps of the ball that are a generalization of parabolic non-automorphism of the unit disk. So he establishes a normal form of these maps (Theorem 3.1) and gives some applications.