Operator Semigroups, Invariant Sets and Invariant Subspaces

Abstract

Let B(H) be the algebra of bounded operators on a complex, separable Hilbert space H. Let (B(H))₁ = {t ∈ B(H): ∥t∥ ≤ 1}. We will say Σ is an operator semigroup if Σ is a unital, absoultely convex subsemigroup of (B(H))₁. In particular, if A is a unital subalgebra of B(H), then A ₁ = {a ∈ B(H): ∥a∥ ≤ 1} is an operator semigroup. If Σ is an operator semigroup, then Σ has a natural action on H ₁, the unit ball of H, namely σ(s, x) = sx, s ∈ Σ, x ∈ H. Consequently, (Σ, H ₁, σ) is a flow when H ₁ is endowed with the weak topology [1]. We apply ideas from the theory of dynamical systems to study operator semigroups, guided by a close analogy between the invariant sets of an operator semigroup Σ and the invariant subspaces of an operator algebra A In particular, we will exhibit invariant set versions of the transitive algebra problem and Lomonosov’s Theorem. There are also interesting connections between certain invariant sets of A ₁ and the hyperinvariant subspaces of A Finally, we study norm precompact orbits for various operator algebras, including self-adjoint algebras, triangular algebras, CSL algebras, and the standard function algebras acting on an L ²space. We summarize some of our results below. Full details will appear in [3]