  We show that the open unit ball of the space of operators from a finite
dimensional Hilbert space into a separable Hilbert space (we call it "operator
ball") has a restricted form of normal structure if we endow it with a
hyperbolic metric (which is an analogue of the standard hyperbolic metric on
the unit disc in the complex plane). We use this result to get a fixed point
theorem for groups of biholomorphic automorphisms of the operator ball. The
fixed point theorem is used to show that a bounded representation in a
separable Hilbert space which has an invariant indefinite quadratic form with
finitely many negative squares is unitarizable (equivalent to a unitary
representation). We apply this result to find dual pairs of invariant subspaces
in Pontryagin spaces. In the appendix we present results of Itai Shafrir about
hyperbolic metrics on the operator ball.
