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

Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls