Operator Semigroups, Invariant Sets and Invariant Subspaces

John Froelich and Michael Marsalli
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John Froelich: University of Houston, Department of Mathematics
Michael Marsalli: Illinois State University, Department of Mathematics

A chapter in Algebraic Methods in Operator Theory, 1994, pp 10-14 from Springer

Abstract: 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]

Keywords: Invariant Subspace; Operator Algebra; Operator Semigroup; Weak Closure; Subnormal Operator (search for similar items in EconPapers)
Date: 1994
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-0255-4_2

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DOI: 10.1007/978-1-4612-0255-4_2

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