STRUCTURE RESULTS FOR CONTRACTIONS WITH ISOMETRIC
FUNCTIONAL CALCULUS
This second talk will begin with an overview of classical structure
results (that have been obtained as applications of Dual algebra theory
and its developments) for absolutely continuous contractions with
isometric Nagy-Foias functional calculus. Among them:
-
the factorization result proved (independently) by Bercovici
and Chevreau (1987) saying that the canonical sesquilinear map (valued
in the predual of the algebra of bounded analytic functions in the open
unit disk) associated to such contractions is onto
-
the reflexivity of such contractions proved by
Brown-Chevreau (1988)
-
results related to the boundary sets introduced by
Chevreau-Exner-Pearcy (1993).
More recent developments, in particular,
-
mapping theorems for these boundary sets
(Cassier-Chalendar-Chevreau 2000)
-
"lifting" various factorization results of the
Bercovici-Chevreau type to the space of integrable functions on the unit
circle (Chalendar-Esterle 1997-1999 )
will also be presented, as well as some of the (numerous) questions
remaining open in this area.