Operator Semigroups, Flows, and Their Invariant Sets

Abstract

Let B( H) be the bounded operators on a Hubert space H. An operator semi-group Σ is an absolutely convex, unital subsemigroup of the ball of B( H). Such a Σ induces a flow on H 1, the unit ball of H, namely ( s, x) → sx. Let l(Σ) be the lattice of closed invariant sets of this flow. Both the weak and norm topologies on H 1 induce Hausdorff metrics on the hyperspace of closed convex subsets of H 1. We show that various subsets of I(Σ) are closed subspaces of these hyperspaces. There exist nontrivial operator semigroups which have the same invariant sets as the ball of B( H); i.e., the "transitive semigroup problem" has a negative solution. However, we can characterize the ball of B( H) in terms of certain transitivity properties. We obtain an analogue of Lomonosov′s Theorem for operator semigroups which are transitive on the unit sphere of H. Let A be a weakly closed unital subalgebra of B( H), and let A 1 be its unit ball. We show that the existence of "minimal invariant sets" for A yields hyperinvariant subspaces. We characterize various operator algebras for which A 1 is strongly precompact including von Neumann algebras, triangular algebras, CSL algebras, and the standard function algebras acting by multiplication on a L 2 space. This yields structural results for certain algebras generated by a subnormal operator.