“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.
Date :
Time : 15.40
Place: Mathematics Seminar Room SA141
Tea
will be served before the seminar.