Holomorphic isometries of Cartan domains of type one (Q1175696)

Let \(H\) and \(K\) be two complex Hilbert spaces one of which is of finite dimension and let \(B\) be the open unit ball in the space of bounded linear maps from \(H\) to \(K\) (with the usual Banach space norm). Let \(\text{Iso} B\) be the semigroup of holomorphic mappings from \(B\) into \(B\) which are isometries for the Kobayashi metric and \((\text{Iso} B)_ 0\) the stability semigroup for 0. The author shows that every element of \((\text{Iso} B)_ 0\) is the restriction to \(B\) of a linear isometric isomorphism. This result permits a description of \((\text{Iso} B)\) in terms of non-invertible operator valued Moebius transformations.