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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | bdopadj 32101 | 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 32102 | 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 32103 | 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 32104 | 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 32105 | 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 32106 | Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.) |
| ⊢ dom adjℎ ⊆ LinOp | ||
| Theorem | nmopadjlei 32107 | 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 32108 | Lemma for nmopadji 32109. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇) | ||
| Theorem | nmopadji 32109 | 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 32110 | 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 32111 | 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 32112 | 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 32113 | 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 32114 | 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 32115 | 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 32116 | 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 32117 | 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 32118 | 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 32119 | 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 32120 | 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 32121 | 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 32122 | 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 32123 | 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 32124 | 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 32125 | 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 32126 | 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 32127 | 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 32128 | 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 32129* | Value of the converse of the bra function. Based on the Riesz Lemma riesz4 32083, 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 32130 | 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 32131 | 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 32132 | 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 32133* | 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 32134 | 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 32135 | 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 32136 | 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 32137 | 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 32138 | 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 32139 | 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 32140 | 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 32141* | 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 32142* | 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 32143* | 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 32144 | 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 32145* | 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 32146 | 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 32147 | The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ HrmOp → 𝑇 ≤op 𝑇) | ||
| Theorem | leopsq 32148 | 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 32149 | 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 32150 | 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 32151 | 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 32152 | 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 32153 | 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 32154 | 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 32155 | 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 32156 | 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 32157 | 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 32158 | 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 32159* | 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 32160* | 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 32161* | 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 32162* | 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 32163* | 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 32164* | 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 32165 | A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻) ∈ HrmOp | ||
| Theorem | pjlnopi 32166 | A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻) ∈ LinOp | ||
| Theorem | pjnmopi 32167 | 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 32168 | A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻) ∈ BndLinOp | ||
| Theorem | pjhmop 32169 | A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) ∈ HrmOp) | ||
| Theorem | hmopidmchi 32170 | 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 32171 | 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 32170 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 32172 | 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 32173 | 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 32174 | 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 32175 | 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 32176 | 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 32177 | 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 32178 | 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 32179 | 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 32180 | 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 32181 | 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 32182 | 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 32183 | 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 32184 | 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 32185 | 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 32186 | 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 32187* | 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 32188 | 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 32189 | 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 32190 | 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))) | ||
| Theorem | pjssposi 32191* | Projector ordering can be expressed by the subset relationship between their projection subspaces. (i)<->(iii) of Theorem 29.2 of [Halmos] p. 48. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (∀𝑥 ∈ ℋ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑥) ·ih 𝑥) ↔ 𝐺 ⊆ 𝐻) | ||
| Theorem | pjordi 32192* | The definition of projector ordering in [Halmos] p. 42 is equivalent to the definition of projector ordering in [Beran] p. 110. (We will usually express projector ordering with the even simpler equivalent 𝐺 ⊆ 𝐻; see pjssposi 32191). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (∀𝑥 ∈ ℋ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑥) ·ih 𝑥) ↔ ((projℎ‘𝐺) “ ℋ) ⊆ ((projℎ‘𝐻) “ ℋ)) | ||
| Theorem | pjssdif2i 32193 | The projection subspace of the difference between two projectors. Part 2 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 32191). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐻) −op (projℎ‘𝐺)) = (projℎ‘(𝐻 ∩ (⊥‘𝐺)))) | ||
| Theorem | pjssdif1i 32194 | A necessary and sufficient condition for the difference between two projectors to be a projector. Part 1 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 32191). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.) |
| ⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐻) −op (projℎ‘𝐺)) ∈ ran projℎ) | ||
| Theorem | pjimai 32195 | The image of a projection. Lemma 5 in Daniel Lehmann, "A presentation of Quantum Logic based on an and then connective", https://doi.org/10.48550/arXiv.quant-ph/0701113. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ Sℋ & ⊢ 𝐵 ∈ Cℋ ⇒ ⊢ ((projℎ‘𝐵) “ 𝐴) = ((𝐴 +ℋ (⊥‘𝐵)) ∩ 𝐵) | ||
| Theorem | pjidmcoi 32196 | A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((projℎ‘𝐻) ∘ (projℎ‘𝐻)) = (projℎ‘𝐻) | ||
| Theorem | pjoccoi 32197 | Composition of projections of a subspace and its orthocomplement. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((projℎ‘𝐻) ∘ (projℎ‘(⊥‘𝐻))) = 0hop | ||
| Theorem | pjtoi 32198 | Subspace sum of projection and projection of orthocomplement. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((projℎ‘𝐻) +op (projℎ‘(⊥‘𝐻))) = (projℎ‘ ℋ) | ||
| Theorem | pjoci 32199 | Projection of orthocomplement. First part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((projℎ‘ ℋ) −op (projℎ‘𝐻)) = (projℎ‘(⊥‘𝐻)) | ||
| Theorem | pjidmco 32200 | A projection operator is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Cℋ → ((projℎ‘𝐻) ∘ (projℎ‘𝐻)) = (projℎ‘𝐻)) | ||
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