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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | lnopl 32001 | Basic linearity property of a linear Hilbert space operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
| ⊢ (((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶))) | ||
| Theorem | unop 32002 | Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵)) | ||
| Theorem | unopf1o 32003 | A unitary operator in Hilbert space is one-to-one and onto. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ) | ||
| Theorem | unopnorm 32004 | A unitary operator is idempotent in the norm. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) = (normℎ‘𝐴)) | ||
| Theorem | cnvunop 32005 | The inverse (converse) of a unitary operator in Hilbert space is unitary. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ UniOp → ◡𝑇 ∈ UniOp) | ||
| Theorem | unopadj 32006 | The inverse (converse) of a unitary operator is its adjoint. Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐴 ·ih (◡𝑇‘𝐵))) | ||
| Theorem | unoplin 32007 | A unitary operator is linear. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp) | ||
| Theorem | counop 32008 | The composition of two unitary operators is unitary. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → (𝑆 ∘ 𝑇) ∈ UniOp) | ||
| Theorem | hmop 32009 | Basic inner product property of a Hermitian operator. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵)) | ||
| Theorem | hmopre 32010 | The inner product of the value and argument of a Hermitian operator is real. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐴) ∈ ℝ) | ||
| Theorem | nmfnlb 32011 | A lower bound for a functional norm. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ≤ (normfn‘𝑇)) | ||
| Theorem | nmfnleub 32012* | An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (Revised by Mario Carneiro, 7-Sep-2014.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴))) | ||
| Theorem | nmfnleub2 32013* | An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → (normfn‘𝑇) ≤ 𝐴) | ||
| Theorem | nmfnge0 32014 | The norm of any Hilbert space functional is nonnegative. (Contributed by NM, 24-May-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ℂ → 0 ≤ (normfn‘𝑇)) | ||
| Theorem | elnlfn 32015 | Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0))) | ||
| Theorem | elnlfn2 32016 | Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ (null‘𝑇)) → (𝑇‘𝐴) = 0) | ||
| Theorem | cnfnc 32017* | Basic continuity property of a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ ContFn ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝐵)) | ||
| Theorem | lnfnl 32018 | Basic linearity property of a linear functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ (((𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶))) | ||
| Theorem | adjcl 32019 | Closure of the adjoint of a Hilbert space operator. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐴) ∈ ℋ) | ||
| Theorem | adj1 32020 | Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵)) | ||
| Theorem | adj2 32021 | Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((adjℎ‘𝑇)‘𝐵))) | ||
| Theorem | adjeq 32022* | A property that determines the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆‘𝑦))) → (adjℎ‘𝑇) = 𝑆) | ||
| Theorem | adjadj 32023 | Double adjoint. Theorem 3.11(iv) of [Beran] p. 106. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘(adjℎ‘𝑇)) = 𝑇) | ||
| Theorem | adjvalval 32024* | Value of the value of the adjoint function. (Contributed by NM, 22-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐴) = (℩𝑤 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤))) | ||
| Theorem | unopadj2 32025 | The adjoint of a unitary operator is its inverse (converse). Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 23-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ UniOp → (adjℎ‘𝑇) = ◡𝑇) | ||
| Theorem | hmopadj 32026 | A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ HrmOp → (adjℎ‘𝑇) = 𝑇) | ||
| Theorem | hmdmadj 32027 | Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ dom adjℎ) | ||
| Theorem | hmopadj2 32028 | An operator is Hermitian iff it is self-adjoint. Definition of Hermitian in [Halmos] p. 41. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ dom adjℎ → (𝑇 ∈ HrmOp ↔ (adjℎ‘𝑇) = 𝑇)) | ||
| Theorem | hmoplin 32029 | A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp) | ||
| Theorem | brafval 32030* | The bra of a vector, expressed as ⟨𝐴 ∣ in Dirac notation. See df-bra 31937. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))) | ||
| Theorem | braval 32031 | A bra-ket juxtaposition, expressed as ⟨𝐴 ∣ 𝐵⟩ in Dirac notation, equals the inner product of the vectors. Based on definition of bra in [Prugovecki] p. 186. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴)) | ||
| Theorem | braadd 32032 | Linearity property of bra for addition. (Contributed by NM, 23-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 +ℎ 𝐶)) = (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶))) | ||
| Theorem | bramul 32033 | Linearity property of bra for multiplication. (Contributed by NM, 23-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 ·ℎ 𝐶)) = (𝐵 · ((bra‘𝐴)‘𝐶))) | ||
| Theorem | brafn 32034 | The bra function is a functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ) | ||
| Theorem | bralnfn 32035 | The Dirac bra function is a linear functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (bra‘𝐴) ∈ LinFn) | ||
| Theorem | bracl 32036 | Closure of the bra function. (Contributed by NM, 23-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) ∈ ℂ) | ||
| Theorem | bra0 32037 | The Dirac bra of the zero vector. (Contributed by NM, 25-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
| ⊢ (bra‘0ℎ) = ( ℋ × {0}) | ||
| Theorem | brafnmul 32038 | Anti-linearity property of bra functional for multiplication. (Contributed by NM, 31-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (bra‘(𝐴 ·ℎ 𝐵)) = ((∗‘𝐴) ·fn (bra‘𝐵))) | ||
| Theorem | kbfval 32039* | The outer product of two vectors, expressed as ∣ 𝐴⟩⟨𝐵 ∣ in Dirac notation. See df-kb 31938. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴))) | ||
| Theorem | kbop 32040 | The outer product of two vectors, expressed as ∣ 𝐴⟩⟨𝐵 ∣ in Dirac notation, is an operator. (Contributed by NM, 30-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵): ℋ⟶ ℋ) | ||
| Theorem | kbval 32041 | The value of the operator resulting from the outer product ∣ 𝐴⟩ ⟨𝐵 ∣ of two vectors. Equation 8.1 of [Prugovecki] p. 376. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴)) | ||
| Theorem | kbmul 32042 | Multiplication property of outer product. (Contributed by NM, 31-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) ketbra 𝐶) = (𝐵 ketbra ((∗‘𝐴) ·ℎ 𝐶))) | ||
| Theorem | kbpj 32043 | If a vector 𝐴 has norm 1, the outer product ∣ 𝐴⟩⟨𝐴 ∣ is the projector onto the subspace spanned by 𝐴. http://en.wikipedia.org/wiki/Bra-ket#Linear%5Foperators. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) → (𝐴 ketbra 𝐴) = (projℎ‘(span‘{𝐴}))) | ||
| Theorem | eleigvec 32044* | Membership in the set of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))) | ||
| Theorem | eleigvec2 32045 | Membership in the set of eigenvectors of a Hilbert space operator. (Contributed by NM, 18-Mar-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ (𝑇‘𝐴) ∈ (span‘{𝐴})))) | ||
| Theorem | eleigveccl 32046 | Closure of an eigenvector of a Hilbert space operator. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → 𝐴 ∈ ℋ) | ||
| Theorem | eigvalval 32047 | The eigenvalue of an eigenvector of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) = (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2))) | ||
| Theorem | eigvalcl 32048 | An eigenvalue is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) ∈ ℂ) | ||
| Theorem | eigvec1 32049 | Property of an eigenvector. (Contributed by NM, 12-Mar-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((𝑇‘𝐴) = (((eigval‘𝑇)‘𝐴) ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) | ||
| Theorem | eighmre 32050 | The eigenvalues of a Hermitian operator are real. Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) ∈ ℝ) | ||
| Theorem | eighmorth 32051 | Eigenvectors of a Hermitian operator with distinct eigenvalues are orthogonal. Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.) |
| ⊢ (((𝑇 ∈ HrmOp ∧ 𝐴 ∈ (eigvec‘𝑇)) ∧ (𝐵 ∈ (eigvec‘𝑇) ∧ ((eigval‘𝑇)‘𝐴) ≠ ((eigval‘𝑇)‘𝐵))) → (𝐴 ·ih 𝐵) = 0) | ||
| Theorem | nmopnegi 32052 | Value of the norm of the negative of a Hilbert space operator. Unlike nmophmi 32118, the operator does not have to be bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (normop‘(-1 ·op 𝑇)) = (normop‘𝑇) | ||
| Theorem | lnop0 32053 | The value of a linear Hilbert space operator at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp → (𝑇‘0ℎ) = 0ℎ) | ||
| Theorem | lnopmul 32054 | Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐵)) = (𝐴 ·ℎ (𝑇‘𝐵))) | ||
| Theorem | lnopli 32055 | Basic scalar product property of a linear Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶))) | ||
| Theorem | lnopfi 32056 | A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ 𝑇: ℋ⟶ ℋ | ||
| Theorem | lnop0i 32057 | The value of a linear Hilbert space operator at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 11-May-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝑇‘0ℎ) = 0ℎ | ||
| Theorem | lnopaddi 32058 | Additive property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +ℎ 𝐵)) = ((𝑇‘𝐴) +ℎ (𝑇‘𝐵))) | ||
| Theorem | lnopmuli 32059 | Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐵)) = (𝐴 ·ℎ (𝑇‘𝐵))) | ||
| Theorem | lnopaddmuli 32060 | Sum/product property of a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +ℎ (𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) +ℎ (𝐴 ·ℎ (𝑇‘𝐶)))) | ||
| Theorem | lnopsubi 32061 | Subtraction property for a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −ℎ 𝐵)) = ((𝑇‘𝐴) −ℎ (𝑇‘𝐵))) | ||
| Theorem | lnopsubmuli 32062 | Subtraction/product property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 −ℎ (𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) −ℎ (𝐴 ·ℎ (𝑇‘𝐶)))) | ||
| Theorem | lnopmulsubi 32063 | Product/subtraction property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘((𝐴 ·ℎ 𝐵) −ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) −ℎ (𝑇‘𝐶))) | ||
| Theorem | homco2 32064 | Move a scalar product out of a composition of operators. The operator 𝑇 must be linear, unlike homco1 31888 that works for any operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 ∘ (𝐴 ·op 𝑈)) = (𝐴 ·op (𝑇 ∘ 𝑈))) | ||
| Theorem | idunop 32065 | The identity function (restricted to Hilbert space) is a unitary operator. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.) |
| ⊢ ( I ↾ ℋ) ∈ UniOp | ||
| Theorem | 0cnop 32066 | The identically zero function is a continuous Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ 0hop ∈ ContOp | ||
| Theorem | 0cnfn 32067 | The identically zero function is a continuous Hilbert space functional. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ ( ℋ × {0}) ∈ ContFn | ||
| Theorem | idcnop 32068 | The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ ( I ↾ ℋ) ∈ ContOp | ||
| Theorem | idhmop 32069 | The Hilbert space identity operator is a Hermitian operator. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.) |
| ⊢ Iop ∈ HrmOp | ||
| Theorem | 0hmop 32070 | The identically zero function is a Hermitian operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.) |
| ⊢ 0hop ∈ HrmOp | ||
| Theorem | 0lnop 32071 | The identically zero function is a linear Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ 0hop ∈ LinOp | ||
| Theorem | 0lnfn 32072 | The identically zero function is a linear Hilbert space functional. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ ( ℋ × {0}) ∈ LinFn | ||
| Theorem | nmop0 32073 | The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.) |
| ⊢ (normop‘ 0hop ) = 0 | ||
| Theorem | nmfn0 32074 | The norm of the identically zero functional is zero. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
| ⊢ (normfn‘( ℋ × {0})) = 0 | ||
| Theorem | hmopbdoptHIL 32075 | A Hermitian operator is a bounded linear operator (Hellinger-Toeplitz Theorem). (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ BndLinOp) | ||
| Theorem | hoddii 32076 | Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 31867 does not require linearity.) (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑅 ∈ LinOp & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑅 ∘ (𝑆 −op 𝑇)) = ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇)) | ||
| Theorem | hoddi 32077 | Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 31867 does not require linearity.) (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑅 ∈ LinOp ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑅 ∘ (𝑆 −op 𝑇)) = ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))) | ||
| Theorem | nmop0h 32078 | The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need ℋ ≠ 0ℋ in nmopun 32101.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.) |
| ⊢ (( ℋ = 0ℋ ∧ 𝑇: ℋ⟶ ℋ) → (normop‘𝑇) = 0) | ||
| Theorem | idlnop 32079 | The identity function (restricted to Hilbert space) is a linear operator. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.) |
| ⊢ ( I ↾ ℋ) ∈ LinOp | ||
| Theorem | 0bdop 32080 | The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ 0hop ∈ BndLinOp | ||
| Theorem | adj0 32081 | Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.) |
| ⊢ (adjℎ‘ 0hop ) = 0hop | ||
| Theorem | nmlnop0iALT 32082 | A linear operator with a zero norm is identically zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop ) | ||
| Theorem | nmlnop0iHIL 32083 | A linear operator with a zero norm is identically zero. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop ) | ||
| Theorem | nmlnopgt0i 32084 | A linear Hilbert space operator that is not identically zero has a positive norm. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝑇 ≠ 0hop ↔ 0 < (normop‘𝑇)) | ||
| Theorem | nmlnop0 32085 | A linear operator with a zero norm is identically zero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp → ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop )) | ||
| Theorem | nmlnopne0 32086 | A linear operator with a nonzero norm is nonzero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp → ((normop‘𝑇) ≠ 0 ↔ 𝑇 ≠ 0hop )) | ||
| Theorem | lnopmi 32087 | The scalar product of a linear operator is a linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ LinOp) | ||
| Theorem | lnophsi 32088 | The sum of two linear operators is linear. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ LinOp & ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝑆 +op 𝑇) ∈ LinOp | ||
| Theorem | lnophdi 32089 | The difference of two linear operators is linear. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ LinOp & ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝑆 −op 𝑇) ∈ LinOp | ||
| Theorem | lnopcoi 32090 | The composition of two linear operators is linear. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ LinOp & ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝑆 ∘ 𝑇) ∈ LinOp | ||
| Theorem | lnopco0i 32091 | The composition of a linear operator with one whose norm is zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ LinOp & ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((normop‘𝑇) = 0 → (normop‘(𝑆 ∘ 𝑇)) = 0) | ||
| Theorem | lnopeq0lem1 32092 | Lemma for lnopeq0i 32094. Apply the generalized polarization identity polid2i 31244 to the quadratic form ((𝑇‘𝑥), 𝑥). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ ((𝑇‘𝐴) ·ih 𝐵) = (((((𝑇‘(𝐴 +ℎ 𝐵)) ·ih (𝐴 +ℎ 𝐵)) − ((𝑇‘(𝐴 −ℎ 𝐵)) ·ih (𝐴 −ℎ 𝐵))) + (i · (((𝑇‘(𝐴 +ℎ (i ·ℎ 𝐵))) ·ih (𝐴 +ℎ (i ·ℎ 𝐵))) − ((𝑇‘(𝐴 −ℎ (i ·ℎ 𝐵))) ·ih (𝐴 −ℎ (i ·ℎ 𝐵)))))) / 4) | ||
| Theorem | lnopeq0lem2 32093 | Lemma for lnopeq0i 32094. (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) = (((((𝑇‘(𝐴 +ℎ 𝐵)) ·ih (𝐴 +ℎ 𝐵)) − ((𝑇‘(𝐴 −ℎ 𝐵)) ·ih (𝐴 −ℎ 𝐵))) + (i · (((𝑇‘(𝐴 +ℎ (i ·ℎ 𝐵))) ·ih (𝐴 +ℎ (i ·ℎ 𝐵))) − ((𝑇‘(𝐴 −ℎ (i ·ℎ 𝐵))) ·ih (𝐴 −ℎ (i ·ℎ 𝐵)))))) / 4)) | ||
| Theorem | lnopeq0i 32094* | A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 31915 for arbitrary operators, when the operator is linear we need to consider only the values of the quadratic form (𝑇‘𝑥) ·ih 𝑥). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = 0 ↔ 𝑇 = 0hop ) | ||
| Theorem | lnopeqi 32095* | Two linear Hilbert space operators are equal iff their quadratic forms are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑈 ∈ LinOp ⇒ ⊢ (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ 𝑇 = 𝑈) | ||
| Theorem | lnopeq 32096* | Two linear Hilbert space operators are equal iff their quadratic forms are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinOp ∧ 𝑈 ∈ LinOp) → (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ 𝑇 = 𝑈)) | ||
| Theorem | lnopunilem1 32097* | Lemma for lnopunii 32099. (Contributed by NM, 14-May-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ (ℜ‘(𝐶 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝐶 · (𝐴 ·ih 𝐵))) | ||
| Theorem | lnopunilem2 32098* | Lemma for lnopunii 32099. (Contributed by NM, 12-May-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵) | ||
| Theorem | lnopunii 32099* | If a linear operator (whose range is ℋ) is idempotent in the norm, the operator is unitary. Similar to theorem in [AkhiezerGlazman] p. 73. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇: ℋ–onto→ ℋ & ⊢ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) ⇒ ⊢ 𝑇 ∈ UniOp | ||
| Theorem | elunop2 32100* | An operator is unitary iff it is linear, onto, and idempotent in the norm. Similar to theorem in [AkhiezerGlazman] p. 73, and its converse. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ UniOp ↔ (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥))) | ||
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