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

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

2008-11-11

arxiv.org/abs/0811.1759v1

Journal of Functional Analysis, 257 (2009), pp. 2476-2496

Mathematics

Functional Analysis

Scientific paper

10.1016/j.jfa.2009.06.021

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.

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