Generalized contractions of the matrix unit disk (Q1124110)

In a previous paper [ibid. 84, 353-357 (1986; Zbl 0611.51022)] the authors considered Möbius transformations on the complex projective line \(P_ 1(M_ n({\mathbb{C}}))\) that are J-equivalent to contractions. These maps carry the matrix unit disk \(\Delta^-\) \((=\) set of pairs of \(n\times n\) matrices \((P_ 1P_ 2)=:P\) such that \(P_ 1P^*_ 1-P_ 2P^*_ 2<0)\) into itself; it was stated - but not yet proved completely - that they decrease the pseudochordal distance. In this paper, a map T: \(\Delta^-\to \Delta^-\) is called a generalized contraction if there exists a pair of points \(P\in \Delta^-\) and \(Q\in \Delta^+\) (complementary set of \(\Delta^-)\) being inverses of each other such that the points PT and QT are inverses of each other, too. The authors prove: A map T: \(\Delta^-\to \Delta^-\) is a generalized contraction if and only if T is J-equivalent to a contraction (Lemma 1); a map T: \(\Delta^-\to \Delta^-\) need not be a generalized contraction (Example 1); a generalized contraction T: \(\Delta^-\to \Delta^-\) decreases the non-Euclidean distance between points in the matrix unit disk (Theorem 1).