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

TL;DR

This paper establishes a fixed point theorem for groups of biholomorphic automorphisms of the operator ball, leading to new insights into unitarizable representations and invariant subspaces in Pontryagin spaces.

Contribution

It introduces a fixed point theorem for automorphism groups of the operator ball and applies it to unitarizability of bounded representations with indefinite quadratic forms.

Findings

Fixed point theorem for automorphism groups of the operator ball

Unitarizability of bounded representations with finitely many negative squares

Identification of dual pairs of invariant subspaces in Pontryagin spaces

Abstract

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.