Unitary Conj is Self Adjoint
LinearPMap.unitaryConj_isSelfAdjoint
Plain-language statement
Unitary conjugation preserves self-adjointness: if A is a self-adjoint operator on H and u : H ≃ₗᵢ[ℂ] H' is unitary, then u A u⁻¹ is self-adjoint on H'. Symmetry, dense domain, and the two deficiency surjectivities of A all transfer through u.
Exact Lean statement
lemma unitaryConj_isSelfAdjoint (u : H ≃ₗᵢ[ℂ] H') {A : H →ₗ.[ℂ] H} (hA : IsSelfAdjoint A) :
IsSelfAdjoint (A.unitaryConj u)Formal artifact
Lean source
lemma unitaryConj_isSelfAdjoint (u : H ≃ₗᵢ[ℂ] H') {A : H →ₗ.[ℂ] H} (hA : IsSelfAdjoint A) : IsSelfAdjoint (A.unitaryConj u) := by have hrange {z : ℂ} (hz : z.im ≠ 0) : (A.unitaryConj u - z • 1).toFun.range = ⊤ := LinearMap.range_eq_top.mpr (unitaryConj_sub_smul_surjective (IsSelfAdjoint.sub_smul_surjective hA hz)) refine IsSymmetric.isSelfAdjoint_of_range_eq_top (IsFormalAdjoint.unitaryConj (IsSelfAdjoint.isSymmetric hA)) (HasDenseDomain.unitaryConj_dense_domain (IsSelfAdjoint.dense_domain hA)) ?_ ?_ · have hI : A.unitaryConj u + I • 1 = A.unitaryConj u - (-I) • 1 := LinearPMap.ext rfl fun x hf hg => by simp [sub_apply, add_apply, smul_apply, sub_neg_eq_add] rw [hI] exact hrange (by norm_num) · exact hrange (by norm_num)- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/Operators/SpectralTheory/SelfAdjoint.lean:119-131
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