#

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

### by

Prof. Mikhail Ostrovskii

(St. John's University)

__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 : June 29, 2009 (Monday)

Time : 15.40

Place: Mathematics Seminar Room SA141

__Tea
will be served before the seminar. __