| import Mathlib.Analysis.Complex.Basic |
| import Mathlib.Analysis.Normed.Operator.ContinuousLinearMap |
| import Mathlib.Algebra.Algebra.Subalgebra.Basic |
| import Mathlib.Topology.Algebra.Module.Spaces.PointwiseConvergenceCLM |
|
|
| namespace OAI |
|
|
| namespace Invariant233 |
|
|
| variable {H : Type*} [NormedAddCommGroup H] [NormedSpace ℂ H] |
|
|
| def operatorCommutant (T : H →L[ℂ] H) : Subalgebra ℂ (H →L[ℂ] H) := |
| Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H)) |
|
|
| def IsInvariant (T : H →L[ℂ] H) (L : Submodule ℂ H) : Prop := |
| ∀ x ∈ L, T x ∈ L |
|
|
| def IsHyperinvariant (T : H →L[ℂ] H) (L : Submodule ℂ H) : Prop := |
| IsClosed (L : Set H) ∧ ∀ A ∈ operatorCommutant T, IsInvariant A L |
|
|
| def IsTransitive (A : Subalgebra ℂ (H →L[ℂ] H)) : Prop := |
| ∀ L : Submodule ℂ H, IsClosed (L : Set H) → |
| (∀ B ∈ A, IsInvariant B L) → L = ⊥ ∨ L = ⊤ |
|
|
| def HasNoNontrivialHyperinvariantSubspace (T : H →L[ℂ] H) : Prop := |
| ∀ L : Submodule ℂ H, IsHyperinvariant T L → L = ⊥ ∨ L = ⊤ |
|
|
| noncomputable def strongOperatorUnderlying (B : H →Lₚₜ[ℂ] H) : H →L[ℂ] H := |
| (UniformConvergenceCLM.ofFun (RingHom.id ℂ) H {s : Set H | s.Finite}).symm B |
|
|
| def IsStronglyClosed (A : Subalgebra ℂ (H →L[ℂ] H)) : Prop := |
| IsClosed {B : H →Lₚₜ[ℂ] H | strongOperatorUnderlying B ∈ A} |
|
|
| end Invariant233 |
|
|
| end OAI |
|
|
