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9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | 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
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