2008/11/11 by Mikhail I. Ostrovskii, M. I. Ostrovskii, Victor S. Shulman +3
Mathematics · #Algebraic and Geometric Analysis #Ball (mathematics) #Hilbert space #Holomorphic and Operator Theory #Invariant (physics) #Linear subspace #Mathematical analysis #Mathematics #Mathematics and Applications #Pure mathematics #Separable space #Unit sphere #math.FA #math.OA #msc:22D10 #msc:47B50 #msc:47H10
paper · pdf · doi:10.1016/j.jfa.2009.06.021
published as Journal of Functional Analysis, 257 (2009), pp. 2476-2496
arxiv created 2008/11/11 · openalex publication_date 2009/07/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.