PETER Rosenthal - Orbit-reflexivity versus orbit-transitivity

The concept of orbit reflexivity was introduced some years ago by Hadwin, Nordgren, Radjavi and the speaker (J. London Math. Soc. (2) 34(1986), 111-119). A little more generally, a collection of bounded linear operators is said to be orbit-reflexive if it contains every operator which leaves invariant all the closed sets invariant under the collection. This is a perturbation of the standard definition of reflexivity, in which ``set'' is replaced by ``subspace''. Every reflexive collection is a strongly closed algebra; every orbit-reflexive collection is a strongly closed semigroup.

A transitive collection of operators is a collection having only trivial invariant subspaces; an orbit-transitive collection is defined to be one that has only trivial invariant closed sets.

In this talk, a general study of orbit-reflexivity and orbit-transitivity will be proposed. A few preliminary results will be presented, including a sufficient condition (obtained in joint work with Radjavi) that a semigroup have a non-trivial invariant closed set. A number of related open problems will be discussed.