Linear fractional maps of the ball and their composition operators. (English) Zbl 0970.47011

The authors describe and investigate a class of maps of the unit ball \(\mathbb B_N\) of \(\mathbb C^N\) into itself. These maps are called linear fractional maps and they are of the type \[ \varphi(z) = (Az+B)(\langle z,C\rangle +D)^{-1}, \] where \(A\) is an \(N \times N\) matrix, \(B\) and \(C\) are column vectors in \(\mathbb C^N\) and \(D\) is a complex number. The \((N+1) \times (N+1)\) matrix \(m_\varphi\) associated with \(\varphi\) is defined to be \[ m_\varphi = \begin{pmatrix} A & B \\ C^* & D \end{pmatrix} . \] One verifies that \(m_{\varphi_1} m_{\varphi_2} = m_{\varphi_1 \circ \varphi_2}\). Furthermore, it is shown that a linear fractional transformation \(\varphi\) maps \(\mathbb B_N\) into itself, if and only if \(m_\varphi\) is a multiple of a matrix T such that \(J - T^*JT\) is positive semidefinite, where \( J = \left(\begin{smallmatrix} I & 0 \\ \phantom{-}0 & -1 \end{smallmatrix}\right)\), i.e., \(T\) is a “Krein space contraction”. Similarly, the linear fractional transformations that map \(\mathbb B_N\) onto itself are the ones whose associated matrices are multiples of Krein space isometries. These maps equal the ball automorphisms.
The authors show that the linear fractional transformations that map the ball into itself define bounded composition operators on \(H^p(\mathbb B_N)\) and on some weighted Bergman spaces. Also, by use of the above mentioned Krein space description, the authors solve Schroeder’s equation \(f \circ \varphi = \varphi'(0)f\) when \(\varphi\) is a linear fractional self map of the ball fixing 0.


47B33 Linear composition operators
47B38 Linear operators on function spaces (general)
32A30 Other generalizations of function theory of one complex variable
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)