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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | cnlnadjlem6 32001* | Lemma for cnlnadji 32005. 𝐹 is linear. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ 𝐹 ∈ LinOp | ||
| Theorem | cnlnadjlem7 32002* | Lemma for cnlnadji 32005. Helper lemma to show that 𝐹 is continuous. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝐹‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | cnlnadjlem8 32003* | Lemma for cnlnadji 32005. 𝐹 is continuous. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ 𝐹 ∈ ContOp | ||
| Theorem | cnlnadjlem9 32004* | Lemma for cnlnadji 32005. 𝐹 provides an example showing the existence of a continuous linear adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑧) = (𝑥 ·ih (𝑡‘𝑧)) | ||
| Theorem | cnlnadji 32005* | Every continuous linear operator has an adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp ⇒ ⊢ ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦)) | ||
| Theorem | cnlnadjeui 32006* | Every continuous linear operator has a unique adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp ⇒ ⊢ ∃!𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦)) | ||
| Theorem | cnlnadjeu 32007* | Every continuous linear operator has a unique adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → ∃!𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦))) | ||
| Theorem | cnlnadj 32008* | Every continuous linear operator has an adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦))) | ||
| Theorem | cnlnssadj 32009 | Every continuous linear Hilbert space operator has an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ (LinOp ∩ ContOp) ⊆ dom adjℎ | ||
| Theorem | bdopssadj 32010 | Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
| ⊢ BndLinOp ⊆ dom adjℎ | ||
| Theorem | bdopadj 32011 | Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ dom adjℎ) | ||
| Theorem | adjbdln 32012 | The adjoint of a bounded linear operator is a bounded linear operator. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ BndLinOp → (adjℎ‘𝑇) ∈ BndLinOp) | ||
| Theorem | adjbdlnb 32013 | An operator is bounded and linear iff its adjoint is. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ BndLinOp ↔ (adjℎ‘𝑇) ∈ BndLinOp) | ||
| Theorem | adjbd1o 32014 | The mapping of adjoints of bounded linear operators is one-to-one onto. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
| ⊢ (adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp | ||
| Theorem | adjlnop 32015 | The adjoint of an operator is linear. Proposition 1 of [AkhiezerGlazman] p. 80. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ LinOp) | ||
| Theorem | adjsslnop 32016 | Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.) |
| ⊢ dom adjℎ ⊆ LinOp | ||
| Theorem | nmopadjlei 32017 | Property of the norm of an adjoint. Part of proof of Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝐴 ∈ ℋ → (normℎ‘((adjℎ‘𝑇)‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmopadjlem 32018 | Lemma for nmopadji 32019. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇) | ||
| Theorem | nmopadji 32019 | Property of the norm of an adjoint. Theorem 3.11(v) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘(adjℎ‘𝑇)) = (normop‘𝑇) | ||
| Theorem | adjeq0 32020 | An operator is zero iff its adjoint is zero. Theorem 3.11(i) of [Beran] p. 106. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 = 0hop ↔ (adjℎ‘𝑇) = 0hop ) | ||
| Theorem | adjmul 32021 | The adjoint of the scalar product of an operator. Theorem 3.11(ii) of [Beran] p. 106. (Contributed by NM, 21-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝐴 ·op 𝑇)) = ((∗‘𝐴) ·op (adjℎ‘𝑇))) | ||
| Theorem | adjadd 32022 | The adjoint of the sum of two operators. Theorem 3.11(iii) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝑆 +op 𝑇)) = ((adjℎ‘𝑆) +op (adjℎ‘𝑇))) | ||
| Theorem | nmoptrii 32023 | Triangle inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)) | ||
| Theorem | nmopcoi 32024 | Upper bound for the norm of the composition of two bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘(𝑆 ∘ 𝑇)) ≤ ((normop‘𝑆) · (normop‘𝑇)) | ||
| Theorem | bdophsi 32025 | The sum of two bounded linear operators is a bounded linear operator. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝑆 +op 𝑇) ∈ BndLinOp | ||
| Theorem | bdophdi 32026 | The difference between two bounded linear operators is bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝑆 −op 𝑇) ∈ BndLinOp | ||
| Theorem | bdopcoi 32027 | The composition of two bounded linear operators is bounded. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝑆 ∘ 𝑇) ∈ BndLinOp | ||
| Theorem | nmoptri2i 32028 | Triangle-type inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ ((normop‘𝑆) − (normop‘𝑇)) ≤ (normop‘(𝑆 +op 𝑇)) | ||
| Theorem | adjcoi 32029 | The adjoint of a composition of bounded linear operators. Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (adjℎ‘(𝑆 ∘ 𝑇)) = ((adjℎ‘𝑇) ∘ (adjℎ‘𝑆)) | ||
| Theorem | nmopcoadji 32030 | The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘((adjℎ‘𝑇) ∘ 𝑇)) = ((normop‘𝑇)↑2) | ||
| Theorem | nmopcoadj2i 32031 | The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘(𝑇 ∘ (adjℎ‘𝑇))) = ((normop‘𝑇)↑2) | ||
| Theorem | nmopcoadj0i 32032 | An operator composed with its adjoint is zero iff the operator is zero. Theorem 3.11(vii) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ ((𝑇 ∘ (adjℎ‘𝑇)) = 0hop ↔ 𝑇 = 0hop ) | ||
| Theorem | unierri 32033 | If we approximate a chain of unitary transformations (quantum computer gates) 𝐹, 𝐺 by other unitary transformations 𝑆, 𝑇, the error increases at most additively. Equation 4.73 of [NielsenChuang] p. 195. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝐹 ∈ UniOp & ⊢ 𝐺 ∈ UniOp & ⊢ 𝑆 ∈ UniOp & ⊢ 𝑇 ∈ UniOp ⇒ ⊢ (normop‘((𝐹 ∘ 𝐺) −op (𝑆 ∘ 𝑇))) ≤ ((normop‘(𝐹 −op 𝑆)) + (normop‘(𝐺 −op 𝑇))) | ||
| Theorem | branmfn 32034 | The norm of the bra function. (Contributed by NM, 24-May-2006.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴)) | ||
| Theorem | brabn 32035 | The bra of a vector is a bounded functional. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) ∈ ℝ) | ||
| Theorem | rnbra 32036 | The set of bras equals the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
| ⊢ ran bra = (LinFn ∩ ContFn) | ||
| Theorem | bra11 32037 | The bra function maps vectors one-to-one onto the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ bra: ℋ–1-1-onto→(LinFn ∩ ContFn) | ||
| Theorem | bracnln 32038 | A bra is a continuous linear functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) ∈ (LinFn ∩ ContFn)) | ||
| Theorem | cnvbraval 32039* | Value of the converse of the bra function. Based on the Riesz Lemma riesz4 31993, this very important theorem not only justifies the Dirac bra-ket notation, but allows to extract a unique vector from any continuous linear functional from which the functional can be recovered; i.e. a single vector can "store" all of the information contained in any entire continuous linear functional (mapping from ℋ to ℂ). (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ (LinFn ∩ ContFn) → (◡bra‘𝑇) = (℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦))) | ||
| Theorem | cnvbracl 32040 | Closure of the converse of the bra function. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ (LinFn ∩ ContFn) → (◡bra‘𝑇) ∈ ℋ) | ||
| Theorem | cnvbrabra 32041 | The converse bra of the bra of a vector is the vector itself. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (◡bra‘(bra‘𝐴)) = 𝐴) | ||
| Theorem | bracnvbra 32042 | The bra of the converse bra of a continuous linear functional. (Contributed by NM, 31-May-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ (LinFn ∩ ContFn) → (bra‘(◡bra‘𝑇)) = 𝑇) | ||
| Theorem | bracnlnval 32043* | The vector that a continuous linear functional is the bra of. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ (LinFn ∩ ContFn) → 𝑇 = (bra‘(℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦)))) | ||
| Theorem | cnvbramul 32044 | Multiplication property of the converse bra function. (Contributed by NM, 31-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ (LinFn ∩ ContFn)) → (◡bra‘(𝐴 ·fn 𝑇)) = ((∗‘𝐴) ·ℎ (◡bra‘𝑇))) | ||
| Theorem | kbass1 32045 | Dirac bra-ket associative law ( ∣ 𝐴⟩⟨𝐵 ∣ ) ∣ 𝐶⟩ = ∣ 𝐴⟩(⟨𝐵 ∣ 𝐶⟩), i.e., the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since ⟨𝐵 ∣ 𝐶⟩ is a complex number, it is the first argument in the inner product ·ℎ that it is mapped to, although in Dirac notation it is placed after the ket. (Contributed by NM, 15-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = (((bra‘𝐵)‘𝐶) ·ℎ 𝐴)) | ||
| Theorem | kbass2 32046 | Dirac bra-ket associative law (⟨𝐴 ∣ 𝐵⟩)⟨𝐶 ∣ = ⟨𝐴 ∣ ( ∣ 𝐵⟩⟨𝐶 ∣ ), i.e., the juxtaposition of an inner product with a bra equals a ket juxtaposed with an outer product. (Contributed by NM, 23-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((bra‘𝐴)‘𝐵) ·fn (bra‘𝐶)) = ((bra‘𝐴) ∘ (𝐵 ketbra 𝐶))) | ||
| Theorem | kbass3 32047 | Dirac bra-ket associative law ⟨𝐴 ∣ 𝐵⟩⟨𝐶 ∣ 𝐷⟩ = (⟨𝐴 ∣ 𝐵⟩⟨𝐶 ∣ ) ∣ 𝐷⟩. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((bra‘𝐴)‘𝐵) · ((bra‘𝐶)‘𝐷)) = ((((bra‘𝐴)‘𝐵) ·fn (bra‘𝐶))‘𝐷)) | ||
| Theorem | kbass4 32048 | Dirac bra-ket associative law ⟨𝐴 ∣ 𝐵⟩⟨𝐶 ∣ 𝐷⟩ = ⟨𝐴 ∣ ( ∣ 𝐵⟩⟨𝐶 ∣ 𝐷⟩). (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((bra‘𝐴)‘𝐵) · ((bra‘𝐶)‘𝐷)) = ((bra‘𝐴)‘(((bra‘𝐶)‘𝐷) ·ℎ 𝐵))) | ||
| Theorem | kbass5 32049 | Dirac bra-ket associative law ( ∣ 𝐴⟩⟨𝐵 ∣ )( ∣ 𝐶⟩⟨𝐷 ∣ ) = (( ∣ 𝐴⟩⟨𝐵 ∣ ) ∣ 𝐶⟩)⟨𝐷 ∣. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷)) | ||
| Theorem | kbass6 32050 | Dirac bra-ket associative law ( ∣ 𝐴⟩⟨𝐵 ∣ )( ∣ 𝐶⟩⟨𝐷 ∣ ) = ∣ 𝐴⟩(⟨𝐵 ∣ ( ∣ 𝐶⟩⟨𝐷 ∣ )). (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (𝐴 ketbra (◡bra‘((bra‘𝐵) ∘ (𝐶 ketbra 𝐷))))) | ||
| Theorem | leopg 32051* | Ordering relation for positive operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐵) → (𝑇 ≤op 𝑈 ↔ ((𝑈 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥)))) | ||
| Theorem | leop 32052* | Ordering relation for operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥))) | ||
| Theorem | leop2 32053* | Ordering relation for operators. Definition of operator ordering in [Young] p. 141. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) | ||
| Theorem | leop3 32054 | Operator ordering in terms of a positive operator. Definition of operator ordering in [Retherford] p. 49. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ 0hop ≤op (𝑈 −op 𝑇))) | ||
| Theorem | leoppos 32055* | Binary relation defining a positive operator. Definition VI.1 of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ HrmOp → ( 0hop ≤op 𝑇 ↔ ∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥))) | ||
| Theorem | leoprf2 32056 | The ordering relation for operators is reflexive. (Contributed by NM, 24-Jul-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 ≤op 𝑇) | ||
| Theorem | leoprf 32057 | The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ HrmOp → 𝑇 ≤op 𝑇) | ||
| Theorem | leopsq 32058 | The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ HrmOp → 0hop ≤op (𝑇 ∘ 𝑇)) | ||
| Theorem | 0leop 32059 | The zero operator is a positive operator. (The literature calls it "positive", even though in some sense it is really "nonnegative".) Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ 0hop ≤op 0hop | ||
| Theorem | idleop 32060 | The identity operator is a positive operator. Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ 0hop ≤op Iop | ||
| Theorem | leopadd 32061 | The sum of two positive operators is positive. Exercise 1(i) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) |
| ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ ( 0hop ≤op 𝑇 ∧ 0hop ≤op 𝑈)) → 0hop ≤op (𝑇 +op 𝑈)) | ||
| Theorem | leopmuli 32062 | The scalar product of a nonnegative real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 0hop ≤op 𝑇)) → 0hop ≤op (𝐴 ·op 𝑇)) | ||
| Theorem | leopmul 32063 | The scalar product of a positive real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 0 < 𝐴) → ( 0hop ≤op 𝑇 ↔ 0hop ≤op (𝐴 ·op 𝑇))) | ||
| Theorem | leopmul2i 32064 | Scalar product applied to operator ordering. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 𝑇 ≤op 𝑈)) → (𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈)) | ||
| Theorem | leoptri 32065 | The positive operator ordering relation satisfies trichotomy. Exercise 1(iii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ((𝑇 ≤op 𝑈 ∧ 𝑈 ≤op 𝑇) ↔ 𝑇 = 𝑈)) | ||
| Theorem | leoptr 32066 | The positive operator ordering relation is transitive. Exercise 1(iv) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) |
| ⊢ (((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑆 ≤op 𝑇 ∧ 𝑇 ≤op 𝑈)) → 𝑆 ≤op 𝑈) | ||
| Theorem | leopnmid 32067 | A bounded Hermitian operator is less than or equal to its norm times the identity operator. (Contributed by NM, 11-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ (normop‘𝑇) ∈ ℝ) → 𝑇 ≤op ((normop‘𝑇) ·op Iop )) | ||
| Theorem | nmopleid 32068 | A nonzero, bounded Hermitian operator divided by its norm is less than or equal to the identity operator. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ (normop‘𝑇) ∈ ℝ ∧ 𝑇 ≠ 0hop ) → ((1 / (normop‘𝑇)) ·op 𝑇) ≤op Iop ) | ||
| Theorem | opsqrlem1 32069* | Lemma for opsqri . (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ HrmOp & ⊢ (normop‘𝑇) ∈ ℝ & ⊢ 0hop ≤op 𝑇 & ⊢ 𝑅 = ((1 / (normop‘𝑇)) ·op 𝑇) & ⊢ (𝑇 ≠ 0hop → ∃𝑢 ∈ HrmOp ( 0hop ≤op 𝑢 ∧ (𝑢 ∘ 𝑢) = 𝑅)) ⇒ ⊢ (𝑇 ≠ 0hop → ∃𝑣 ∈ HrmOp ( 0hop ≤op 𝑣 ∧ (𝑣 ∘ 𝑣) = 𝑇)) | ||
| Theorem | opsqrlem2 32070* | Lemma for opsqri . 𝐹‘𝑁 is the recursive function An (starting at n=1 instead of 0) of Theorem 9.4-2 of [Kreyszig] p. 476. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ HrmOp & ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥))))) & ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) ⇒ ⊢ (𝐹‘1) = 0hop | ||
| Theorem | opsqrlem3 32071* | Lemma for opsqri . (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ HrmOp & ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥))))) & ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) ⇒ ⊢ ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺))))) | ||
| Theorem | opsqrlem4 32072* | Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ HrmOp & ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥))))) & ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) ⇒ ⊢ 𝐹:ℕ⟶HrmOp | ||
| Theorem | opsqrlem5 32073* | Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ HrmOp & ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥))))) & ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) ⇒ ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) = ((𝐹‘𝑁) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑁) ∘ (𝐹‘𝑁)))))) | ||
| Theorem | opsqrlem6 32074* | Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ HrmOp & ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥))))) & ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) & ⊢ 𝑇 ≤op Iop ⇒ ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤op Iop ) | ||
| Theorem | pjhmopi 32075 | A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻) ∈ HrmOp | ||
| Theorem | pjlnopi 32076 | A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻) ∈ LinOp | ||
| Theorem | pjnmopi 32077 | The operator norm of a projector on a nonzero closed subspace is one. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐻 ≠ 0ℋ → (normop‘(projℎ‘𝐻)) = 1) | ||
| Theorem | pjbdlni 32078 | A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻) ∈ BndLinOp | ||
| Theorem | pjhmop 32079 | A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) ∈ HrmOp) | ||
| Theorem | hmopidmchi 32080 | An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 21-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ HrmOp & ⊢ (𝑇 ∘ 𝑇) = 𝑇 ⇒ ⊢ ran 𝑇 ∈ Cℋ | ||
| Theorem | hmopidmpji 32081 | An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Halmos seems to omit the proof that 𝐻 is a closed subspace, which is not trivial as hmopidmchi 32080 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ HrmOp & ⊢ (𝑇 ∘ 𝑇) = 𝑇 ⇒ ⊢ 𝑇 = (projℎ‘ran 𝑇) | ||
| Theorem | hmopidmch 32082 | An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ (𝑇 ∘ 𝑇) = 𝑇) → ran 𝑇 ∈ Cℋ ) | ||
| Theorem | hmopidmpj 32083 | An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ (𝑇 ∘ 𝑇) = 𝑇) → 𝑇 = (projℎ‘ran 𝑇)) | ||
| Theorem | pjsdii 32084 | Distributive law for Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((projℎ‘𝐻) ∘ (𝑆 +op 𝑇)) = (((projℎ‘𝐻) ∘ 𝑆) +op ((projℎ‘𝐻) ∘ 𝑇)) | ||
| Theorem | pjddii 32085 | Distributive law for Hilbert space operator difference. (Contributed by NM, 24-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((projℎ‘𝐻) ∘ (𝑆 −op 𝑇)) = (((projℎ‘𝐻) ∘ 𝑆) −op ((projℎ‘𝐻) ∘ 𝑇)) | ||
| Theorem | pjsdi2i 32086 | Chained distributive law for Hilbert space operator difference. (Contributed by NM, 30-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ & ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 ∘ (𝑆 +op 𝑇)) = ((𝑅 ∘ 𝑆) +op (𝑅 ∘ 𝑇)) → (((projℎ‘𝐻) ∘ 𝑅) ∘ (𝑆 +op 𝑇)) = ((((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑆) +op (((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑇))) | ||
| Theorem | pjcoi 32087 | Composition of projections. (Contributed by NM, 16-Aug-2000.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) = ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝐴))) | ||
| Theorem | pjcocli 32088 | Closure of composition of projections. (Contributed by NM, 29-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) ∈ 𝐺) | ||
| Theorem | pjcohcli 32089 | Closure of composition of projections. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) ∈ ℋ) | ||
| Theorem | pjadjcoi 32090 | Adjoint of composition of projections. Special case of Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 6-Oct-2000.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) ·ih 𝐵) = (𝐴 ·ih (((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝐵))) | ||
| Theorem | pjcofni 32091 | Functionality of composition of projections. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) Fn ℋ | ||
| Theorem | pjss1coi 32092 | Subset relationship for projections. Theorem 4.5(i)<->(iii) of [Beran] p. 112. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) = (projℎ‘𝐺)) | ||
| Theorem | pjss2coi 32093 | Subset relationship for projections. Theorem 4.5(i)<->(ii) of [Beran] p. 112. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺)) | ||
| Theorem | pjssmi 32094 | Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → (𝐻 ⊆ 𝐺 → (((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) = ((projℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴))) | ||
| Theorem | pjssge0i 32095 | Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → ((((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) = ((projℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) → 0 ≤ ((((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) ·ih 𝐴))) | ||
| Theorem | pjdifnormi 32096 | Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → (0 ≤ ((((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) ·ih 𝐴) ↔ (normℎ‘((projℎ‘𝐻)‘𝐴)) ≤ (normℎ‘((projℎ‘𝐺)‘𝐴)))) | ||
| Theorem | pjnormssi 32097* | Theorem 4.5(i)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐺 ⊆ 𝐻 ↔ ∀𝑥 ∈ ℋ (normℎ‘((projℎ‘𝐺)‘𝑥)) ≤ (normℎ‘((projℎ‘𝐻)‘𝑥))) | ||
| Theorem | pjorthcoi 32098 | Composition of projections of orthogonal subspaces. Part (i)->(iia) of Theorem 27.4 of [Halmos] p. 45. (Contributed by NM, 6-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐺 ⊆ (⊥‘𝐻) → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = 0hop ) | ||
| Theorem | pjscji 32099 | The projection of orthogonal subspaces is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐺 ⊆ (⊥‘𝐻) → (projℎ‘(𝐺 ∨ℋ 𝐻)) = ((projℎ‘𝐺) +op (projℎ‘𝐻))) | ||
| Theorem | pjssumi 32100 | The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐺 ⊆ (⊥‘𝐻) → (projℎ‘(𝐺 +ℋ 𝐻)) = ((projℎ‘𝐺) +op (projℎ‘𝐻))) | ||
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