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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | hodcli 31901 | Closure of Hilbert space operator difference. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝐴 ∈ ℋ → ((𝑆 −op 𝑇)‘𝐴) ∈ ℋ) | ||
| Theorem | hocoi 31902 | Composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) = (𝑆‘(𝑇‘𝐴))) | ||
| Theorem | hococli 31903 | Closure of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) ∈ ℋ) | ||
| Theorem | hocofi 31904 | Mapping of composition of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ | ||
| Theorem | hocofni 31905 | Functionality of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑆 ∘ 𝑇) Fn ℋ | ||
| Theorem | hoaddcli 31906 | Mapping of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ | ||
| Theorem | hosubcli 31907 | Mapping of difference of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ | ||
| Theorem | hoaddfni 31908 | Functionality of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑆 +op 𝑇) Fn ℋ | ||
| Theorem | hosubfni 31909 | Functionality of difference of Hilbert space operators. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑆 −op 𝑇) Fn ℋ | ||
| Theorem | hoaddcomi 31910 | Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑆 +op 𝑇) = (𝑇 +op 𝑆) | ||
| Theorem | hosubcl 31911 | Mapping of difference of Hilbert space operators. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 −op 𝑇): ℋ⟶ ℋ) | ||
| Theorem | hoaddcom 31912 | Commutativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑇 +op 𝑆)) | ||
| Theorem | hodsi 31913 | Relationship between Hilbert space operator difference and sum. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 −op 𝑆) = 𝑇 ↔ (𝑆 +op 𝑇) = 𝑅) | ||
| Theorem | hoaddassi 31914 | Associativity of sum of Hilbert space operators. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇)) | ||
| Theorem | hoadd12i 31915 | Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.) |
| ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑅 +op (𝑆 +op 𝑇)) = (𝑆 +op (𝑅 +op 𝑇)) | ||
| Theorem | hoadd32i 31916 | Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
| ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆) | ||
| Theorem | hocadddiri 31917 | Distributive law for Hilbert space operator sum. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇)) | ||
| Theorem | hocsubdiri 31918 | Distributive law for Hilbert space operator difference. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 −op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) −op (𝑆 ∘ 𝑇)) | ||
| Theorem | ho2coi 31919 | Double composition of Hilbert space operators. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.) |
| ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝐴 ∈ ℋ → (((𝑅 ∘ 𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇‘𝐴)))) | ||
| Theorem | hoaddass 31920 | Associativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))) | ||
| Theorem | hoadd32 31921 | Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆)) | ||
| Theorem | hoadd4 31922 | Rearrangement of 4 terms in a sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ (((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝑅 +op 𝑆) +op (𝑇 +op 𝑈)) = ((𝑅 +op 𝑇) +op (𝑆 +op 𝑈))) | ||
| Theorem | hocsubdir 31923 | Distributive law for Hilbert space operator difference. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑅 −op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) −op (𝑆 ∘ 𝑇))) | ||
| Theorem | hoaddridi 31924 | Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑇 +op 0hop ) = 𝑇 | ||
| Theorem | hodidi 31925 | Difference of a Hilbert space operator from itself. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑇 −op 𝑇) = 0hop | ||
| Theorem | ho0coi 31926 | Composition of the zero operator and a Hilbert space operator. (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ( 0hop ∘ 𝑇) = 0hop | ||
| Theorem | hoid1i 31927 | Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑇 ∘ Iop ) = 𝑇 | ||
| Theorem | hoid1ri 31928 | Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ( Iop ∘ 𝑇) = 𝑇 | ||
| Theorem | hoaddrid 31929 | Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 +op 0hop ) = 𝑇) | ||
| Theorem | hodid 31930 | Difference of a Hilbert space operator from itself. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 −op 𝑇) = 0hop ) | ||
| Theorem | hon0 31931 | A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅) | ||
| Theorem | hodseqi 31932 | Subtraction and addition of equal Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑆 +op (𝑇 −op 𝑆)) = 𝑇 | ||
| Theorem | ho0subi 31933 | Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑆 −op 𝑇) = (𝑆 +op ( 0hop −op 𝑇)) | ||
| Theorem | honegsubi 31934 | Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑆 +op (-1 ·op 𝑇)) = (𝑆 −op 𝑇) | ||
| Theorem | ho0sub 31935 | Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 −op 𝑇) = (𝑆 +op ( 0hop −op 𝑇))) | ||
| Theorem | hosubid1 31936 | The zero operator subtracted from a Hilbert space operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 −op 0hop ) = 𝑇) | ||
| Theorem | honegsub 31937 | Relationship between Hilbert space operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 +op (-1 ·op 𝑈)) = (𝑇 −op 𝑈)) | ||
| Theorem | homullid 31938 | An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (1 ·op 𝑇) = 𝑇) | ||
| Theorem | homco1 31939 | Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) ∘ 𝑈) = (𝐴 ·op (𝑇 ∘ 𝑈))) | ||
| Theorem | homulass 31940 | Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇))) | ||
| Theorem | hoadddi 31941 | Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)) = ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))) | ||
| Theorem | hoadddir 31942 | Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇) = ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇))) | ||
| Theorem | homul12 31943 | Swap first and second factors in a nested operator scalar product. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op (𝐵 ·op 𝑇)) = (𝐵 ·op (𝐴 ·op 𝑇))) | ||
| Theorem | honegneg 31944 | Double negative of a Hilbert space operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (-1 ·op (-1 ·op 𝑇)) = 𝑇) | ||
| Theorem | hosubneg 31945 | Relationship between operator subtraction and negative. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 −op (-1 ·op 𝑈)) = (𝑇 +op 𝑈)) | ||
| Theorem | hosubdi 31946 | Scalar product distributive law for operator difference. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 −op 𝑈)) = ((𝐴 ·op 𝑇) −op (𝐴 ·op 𝑈))) | ||
| Theorem | honegdi 31947 | Distribution of negative over addition. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇 +op 𝑈)) = ((-1 ·op 𝑇) +op (-1 ·op 𝑈))) | ||
| Theorem | honegsubdi 31948 | Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇 −op 𝑈)) = ((-1 ·op 𝑇) +op 𝑈)) | ||
| Theorem | honegsubdi2 31949 | Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇 −op 𝑈)) = (𝑈 −op 𝑇)) | ||
| Theorem | hosubsub2 31950 | Law for double subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆 −op (𝑇 −op 𝑈)) = (𝑆 +op (𝑈 −op 𝑇))) | ||
| Theorem | hosub4 31951 | Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ (((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝑅 +op 𝑆) −op (𝑇 +op 𝑈)) = ((𝑅 −op 𝑇) +op (𝑆 −op 𝑈))) | ||
| Theorem | hosubadd4 31952 | Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ (((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝑅 −op 𝑆) −op (𝑇 −op 𝑈)) = ((𝑅 +op 𝑈) −op (𝑆 +op 𝑇))) | ||
| Theorem | hoaddsubass 31953 | Associative-type law for addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆 +op 𝑇) −op 𝑈) = (𝑆 +op (𝑇 −op 𝑈))) | ||
| Theorem | hoaddsub 31954 | Law for operator addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆 +op 𝑇) −op 𝑈) = ((𝑆 −op 𝑈) +op 𝑇)) | ||
| Theorem | hosubsub 31955 | Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆 −op (𝑇 −op 𝑈)) = ((𝑆 −op 𝑇) +op 𝑈)) | ||
| Theorem | hosubsub4 31956 | Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆 −op 𝑇) −op 𝑈) = (𝑆 −op (𝑇 +op 𝑈))) | ||
| Theorem | ho2times 31957 | Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = (𝑇 +op 𝑇)) | ||
| Theorem | hoaddsubassi 31958 | Associativity of sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.) |
| ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 +op 𝑆) −op 𝑇) = (𝑅 +op (𝑆 −op 𝑇)) | ||
| Theorem | hoaddsubi 31959 | Law for sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.) |
| ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 +op 𝑆) −op 𝑇) = ((𝑅 −op 𝑇) +op 𝑆) | ||
| Theorem | hosd1i 31960 | Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ & ⊢ 𝑈: ℋ⟶ ℋ ⇒ ⊢ (𝑇 +op 𝑈) = (𝑇 −op ( 0hop −op 𝑈)) | ||
| Theorem | hosd2i 31961 | Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ & ⊢ 𝑈: ℋ⟶ ℋ ⇒ ⊢ (𝑇 +op 𝑈) = (𝑇 −op ((𝑈 −op 𝑈) −op 𝑈)) | ||
| Theorem | hopncani 31962 | Hilbert space operator cancellation law. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ & ⊢ 𝑈: ℋ⟶ ℋ ⇒ ⊢ ((𝑇 +op 𝑈) −op 𝑈) = 𝑇 | ||
| Theorem | honpcani 31963 | Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ & ⊢ 𝑈: ℋ⟶ ℋ ⇒ ⊢ ((𝑇 −op 𝑈) +op 𝑈) = 𝑇 | ||
| Theorem | hosubeq0i 31964 | If the difference between two operators is zero, they are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ & ⊢ 𝑈: ℋ⟶ ℋ ⇒ ⊢ ((𝑇 −op 𝑈) = 0hop ↔ 𝑇 = 𝑈) | ||
| Theorem | honpncani 31965 | Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 −op 𝑆) +op (𝑆 −op 𝑇)) = (𝑅 −op 𝑇) | ||
| Theorem | ho01i 31966* | A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S8) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = 0 ↔ 𝑇 = 0hop ) | ||
| Theorem | ho02i 31967* | A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S10) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = 0 ↔ 𝑇 = 0hop ) | ||
| Theorem | hoeq1 31968* | A condition implying that two Hilbert space operators are equal. Lemma 3.2(S9) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑦) = ((𝑇‘𝑥) ·ih 𝑦) ↔ 𝑆 = 𝑇)) | ||
| Theorem | hoeq2 31969* | A condition implying that two Hilbert space operators are equal. Lemma 3.2(S11) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆‘𝑦)) = (𝑥 ·ih (𝑇‘𝑦)) ↔ 𝑆 = 𝑇)) | ||
| Theorem | adjmo 31970* | Every Hilbert space operator has at most one adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ ∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦)) | ||
| Theorem | adjsym 31971* | Symmetry property of an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·ih 𝑦))) | ||
| Theorem | eigrei 31972 | A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for an eigenvalue 𝐵 to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 21-Jan-2005.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ)) | ||
| Theorem | eigre 31973 | A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for an eigenvalue 𝐵 to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ)) | ||
| Theorem | eigposi 31974 | A sufficient condition (first conjunct pair, that holds when 𝑇 is a positive operator) for an eigenvalue 𝐵 (second conjunct pair) to be nonnegative. Remark (ii) in [Hughes] p. 137. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) | ||
| Theorem | eigorthi 31975 | A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for two eigenvectors 𝐴 and 𝐵 to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Jan-2005.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ ⇒ ⊢ ((((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0)) | ||
| Theorem | eigorth 31976 | A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for two eigenvectors 𝐴 and 𝐵 to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.) |
| ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) ∧ (((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))) → ((𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0)) | ||
| Definition | df-nmop 31977* | Define the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.) |
| ⊢ normop = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑧)))}, ℝ*, < )) | ||
| Definition | df-cnop 31978* | Define the set of continuous operators on Hilbert space. For every "epsilon" (𝑦) there is a "delta" (𝑧) such that... (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.) |
| ⊢ ContOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑡‘𝑤) −ℎ (𝑡‘𝑥))) < 𝑦)} | ||
| Definition | df-lnop 31979* | Define the set of linear operators on Hilbert space. (See df-hosum 31868 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.) |
| ⊢ LinOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑡‘𝑦)) +ℎ (𝑡‘𝑧))} | ||
| Definition | df-bdop 31980 | Define the set of bounded linear Hilbert space operators. (See df-hosum 31868 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.) |
| ⊢ BndLinOp = {𝑡 ∈ LinOp ∣ (normop‘𝑡) < +∞} | ||
| Definition | df-unop 31981* | Define the set of unitary operators on Hilbert space. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.) |
| ⊢ UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦))} | ||
| Definition | df-hmop 31982* | Define the set of Hermitian operators on Hilbert space. Some books call these "symmetric operators" and others call them "self-adjoint operators", sometimes with slightly different technical meanings. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.) |
| ⊢ HrmOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·ih 𝑦)} | ||
| Definition | df-nmfn 31983* | Define the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ normfn = (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, < )) | ||
| Definition | df-nlfn 31984 | Define the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ null = (𝑡 ∈ (ℂ ↑m ℋ) ↦ (◡𝑡 “ {0})) | ||
| Definition | df-cnfn 31985* | Define the set of continuous functionals on Hilbert space. For every "epsilon" (𝑦) there is a "delta" (𝑧) such that... (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ ContFn = {𝑡 ∈ (ℂ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)} | ||
| Definition | df-lnfn 31986* | Define the set of linear functionals on Hilbert space. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ LinFn = {𝑡 ∈ (ℂ ↑m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧))} | ||
| Definition | df-adjh 31987* | Define the adjoint of a Hilbert space operator (if it exists). The domain of adjℎ is the set of all adjoint operators. Definition of adjoint in [Kalmbach2] p. 8. Unlike Kalmbach (and most authors), we do not demand that the operator be linear, but instead show (in adjbdln 32221) that the adjoint exists for a bounded linear operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.) |
| ⊢ adjℎ = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑢‘𝑦)))} | ||
| Definition | df-bra 31988* |
Define the bra of a vector used by Dirac notation. Based on definition
of bra in [Prugovecki] p. 186 (p.
180 in 1971 edition). In Dirac
bra-ket notation, ⟨𝐴 ∣ 𝐵⟩ is a complex number equal to
the inner
product (𝐵 ·ih 𝐴). But physicists like
to talk about the
individual components ⟨𝐴 ∣ and ∣
𝐵⟩, called bra
and ket
respectively. In order for their properties to make sense formally, we
define the ket ∣ 𝐵⟩ as the vector 𝐵 itself,
and the bra
⟨𝐴 ∣ as a functional from ℋ to ℂ. We represent the
Dirac notation ⟨𝐴 ∣ 𝐵⟩ by ((bra‘𝐴)‘𝐵); see
braval 32082. The reversal of the inner product
arguments not only makes
the bra-ket behavior consistent with physics literature (see comments
under ax-his3 31222) but is also required in order for the
associative law
kbass2 32255 to work.
Our definition of bra and the associated outer product df-kb 31989 differs from, but is equivalent to, a common approach in the literature that makes use of mappings to a dual space. Our approach eliminates the need to have a parallel development of this dual space and instead keeps everything in Hilbert space. For an extensive discussion about how our notation maps to the bra-ket notation in physics textbooks, see mmnotes.txt 31989, under the 17-May-2006 entry. (Contributed by NM, 15-May-2006.) (New usage is discouraged.) |
| ⊢ bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·ih 𝑥))) | ||
| Definition | df-kb 31989* | Define a commuted bra and ket juxtaposition used by Dirac notation. In Dirac notation, ∣ 𝐴⟩⟨𝐵 ∣ is an operator known as the outer product of 𝐴 and 𝐵, which we represent by (𝐴 ketbra 𝐵). Based on Equation 8.1 of [Prugovecki] p. 376. This definition, combined with Definition df-bra 31988, allows any legal juxtaposition of bras and kets to make sense formally and also to obey the associative law when mapped back to Dirac notation. (Contributed by NM, 15-May-2006.) (New usage is discouraged.) |
| ⊢ ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥))) | ||
| Definition | df-leop 31990* | Define positive operator ordering. Definition VI.1 of [Retherford] p. 49. Note that ( ℋ × 0ℋ) ≤op 𝑇 means that 𝑇 is a positive operator. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ ≤op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢 −op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥))} | ||
| Definition | df-eigvec 31991* | Define the eigenvector function. Theorem eleigveccl 32097 shows that eigvec‘𝑇, the set of eigenvectors of Hilbert space operator 𝑇, are Hilbert space vectors. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
| ⊢ eigvec = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·ℎ 𝑥)}) | ||
| Definition | df-eigval 31992* | Define the eigenvalue function. The range of eigval‘𝑇 is the set of eigenvalues of Hilbert space operator 𝑇. Theorem eigvalcl 32099 shows that (eigval‘𝑇)‘𝐴, the eigenvalue associated with eigenvector 𝐴, is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
| ⊢ eigval = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))) | ||
| Definition | df-spec 31993* | Define the spectrum of an operator. Definition of spectrum in [Halmos] p. 50. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.) |
| ⊢ Lambda = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ}) | ||
| Theorem | nmopval 31994* | Value of the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < )) | ||
| Theorem | elcnop 31995* | Property defining a continuous Hilbert space operator. (Contributed by NM, 28-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ ContOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))) < 𝑦))) | ||
| Theorem | ellnop 31996* | Property defining a linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))) | ||
| Theorem | lnopf 31997 | A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ) | ||
| Theorem | elbdop 31998 | Property defining a bounded linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞)) | ||
| Theorem | bdopln 31999 | A bounded linear Hilbert space operator is a linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp) | ||
| Theorem | bdopf 32000 | A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ) | ||
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