“Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls”
Prof. Mikhail Ostrovskii
Abstract: We show that a bounded representation of a group by operators in a Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). To achieve this goal we use an observation (going back to M.~Krein) that operators leaving invariant such indefinite quadratic form correspond to biholomorphic maps of the open unit operator ball B (from a finite-dimensional Hilbert space into an infinite-dimensional). It is not difficult to verify that to get unitarizability we need to show that the obtained group of biholomorphic mappings has a fixed point. To get a fixed point we use such well-known tools as non-diametral points, ball-compactness, and the normal structure. We show that B with Caratheodory metric is ball-compact and has the normal structure.
Time : 15.40
Place: Mathematics Seminar Room SA141
Tea will be served before the seminar.