Invariant subspaces for commuting contractions
by
Joerg Eschmeier
Fachbereich Mathematik, Universitaet des Saarlandes, Germany

A classical result of Brown, Chevreau, and Pearcy from 1979 says that each contraction T on a complex Hilbert space H such that the spectrum of T is dominating in the open unit disc possesses a non-trivial closed invariant subspace. This was the starting point for a series of invariant-subspace and reflexivity results for single contractions with rich spectrum or isometric H^\infty-functional calculus. In 1988 Brown, Chevreau, and Pearcy proved that each contraction T on H whose spectrum contains the unit circle possesses a non-trivial invariant subspace. In the same year Brown and Chevreau showed that each contraction with an isometric H^\infty-functional calculus is reflexive.

In the planned talk we consider commuting systems T = (T₁, ... , T_n) of contractions over the unit ball or the unit polydisc and indicate how a multivariable version of the Scott Brown technique can be used to prove results on the existence of joint invariant subspaces or the reflexivity of suitable commuting families of contractions. Among other results we obtain that a commuting n-tuple of contractions that possesses a unitary dilation and whose Harte spectrum is dominating in the unit polydisc possesses a non-trivial joint invariant subspace. The same result holds for commuting contractions with a spherical dilation and rich joint spectrum in the unit ball. As applications we prove the reflexivity of subnormal n-tuples with rich Taylor spectrum or isometric H^\infty-functional calculus over the unit ball or the unit polydisc.

To prove our results we relate the study of norm-continuous representations of the ball or polydisc algebra to the theory of Henkin measures and we use the abstract F.and M.Riesz theorem to decompose such representations into an absolutely continuous and a singular part. Extending results of Sz.-Nagy and Foias we show that the ball algebra functional calculus of a completely non-unitary spherical contraction is absolutely continuous and that a non-scalar spherical contraction for which the ball algebra functional calculus and its adjoint both are not weak-*-SOT continuous possesses a non-trivial hyperinvariant subspace.

These results together with the Scott Brown factorization technique are the main ingredients used to prove the above mentioned multivariable invariant subspace and reflexivity results.

[1] J.Eschmeier, Invariant subspaces for spherical contractions, Proc.London Math. Soc.75(1997), 157-176

[2] J.Eschmeier, Algebras of subnormal operators on the unit ball, J.Operator Theory 42(1999), 37-76

[3] J.Eschmeier, On the structure of spherical contractions, to appear.

(T)

Date received: April 14, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caey-53.