The Schwarz-Pick Theorem for the Unit Disk of the Projective Matrix Space

The one-dimensional left-projective space over the complex n × n matrices P = P1 (Mn(C)) was considered by B. Schwarz and A. Zaks (J. Algebra95, 1985. 263-307). The projectivities T of P onto itself, the Euclidean distance d(P, Q) between finite points P and Q, and the non-Euclidean distance En(P, Q) between points in the open unit disk Δ− of P were studied in detail. Here we prove the following assertions about projectivities mapping Δ− into itself or into its closure Δ−. (O denotes the origin of P, i.e., the center of Δ−.) LEMMA. If T(Δ−) ⊂ Δ−and T(O) = O, then d(O,T(P)) ≤ d(O, P) for every pointPin Δ−. THEOREM. If T(Δ−) ⊂ Δ−, then En(T(P), T(Q)) ≤ En(P, Q) for every pair of points in Δ−. For a recent generalizatlon of this result see Theorem 4.1 of B. Schwarz (Analysis10, 1990, 319-335).