Is Unbounded orthogonal adjoint sub ker
LinearPMap.IsUnbounded.orthogonal_adjoint_sub_ker
Plain-language statement
(T† - conj z • 1).kerᗮ = (T.closure - z • 1).range
Exact Lean statement
lemma IsUnbounded.orthogonal_adjoint_sub_ker [CompleteSpace H]
{T : H →ₗ.[ℂ] H} (hT : T.IsUnbounded) {z : ℂ} (hz : z ∈ T.regularityDomain) :
((T† - conj z • 1).toFun.ker.map (T† - conj z • 1).domain.subtype)ᗮ
= (T.closure - z • 1).toFun.rangeFormal artifact
Lean source
lemma IsUnbounded.orthogonal_adjoint_sub_ker [CompleteSpace H] {T : H →ₗ.[ℂ] H} (hT : T.IsUnbounded) {z : ℂ} (hz : z ∈ T.regularityDomain) : ((T† - conj z • 1).toFun.ker.map (T† - conj z • 1).domain.subtype)ᗮ = (T.closure - z • 1).toFun.range := by have hT' : IsClosable T.closure := hT.isClosable.closureIsClosable have hTcl : T.closure.closure = T.closure := hT.isClosable.closure_isClosed.closure_eq rw [← hTcl, ← hT.orthogonal_closure_sub_range, orthogonal_orthogonal_eq_closure] exact hT'.closure_range_sub_eq_range_closure_sub (T.regularityDomain_closure ▸ hz)- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/Operators/SpectralTheory/Basic.lean:294-301
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