**Date & Time:** Tuesday, January 06, 2009, 14:30-15:30.

**Venue:** Ramanujan Hall

**Title:** On the Reflexivity of Finite Dimensional Algebras and Subspaces of Operators

**Speaker:** Marek Ptak, University of Agriculture in Krakow

**Abstract:** The algebra of operators on the Hilbert space is called reflexive if there are so many invariant subspaces for all operators from the algebra that they determine the algebra itself. We present the reflexivity results starting with the situation when underlying Hilbert space is finite dimensional and giving some examples showing that even in this case the notion of reflexivity is interesting. We especially concentrate on the algebra generated by a single nilpotent and by a pair of doubly commuting nilpotents. The notion of reflexivity can be naturally extended from algebras to subspaces. There is also stronger notion hyperreflexivity. The algebra (the subspace) is hyperreflexive if the usual distance from any operator to the algebra (subspace) can be controlled by the distance given by invariant subspaces(rank one operators from the preannihilator). It will be presented that reflexivity and hyperreflexivity are equivalent for finite dimensional subspaces of operators even the underlying Hilbert space is not finite dimensional (positive answer for LarsonĀKraus problem).