ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 47 Ptak, Marek, Kraków, Poland Toeplitz operators; dichotomy of reflexivity and transitivity Let S be a linear manifold of bounded Hilbert space operators. An operator A belongs to ref S, the reflexive closure of S if Af belongs to the closure of Sf for each vector f in the underlying Hilbert space. Two extreme possibilities are (1) S is reflexive in the sense that ref S = S, and (2) S is transitive in the sense that ref S includes all bounded operators on the underlying space. We show that every linear space B of Toeplitz operators which is closed in the ultraweak operator topology is either transitive or reflexive. No intermediate behavior is possible. The results cover joint work with E. A. Azoff.