Linear Operators, Part 2 |
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Page 958
... disjoint . Thus y ( e1 ) and ( e ) are orthogonal whenever e , and e , are disjoint . Hence if e , and e are disjoint then y ( e1 e2 ) = E ( e1 U е2 ) y ( е1 U е2 ) = = [ E ( e1 ) + E ( e2 ) ] ¥ ( e1 ~ € 2 ) E ( e1 ) y ( e1 U е2 ) + E ...
... disjoint . Thus y ( e1 ) and ( e ) are orthogonal whenever e , and e , are disjoint . Hence if e , and e are disjoint then y ( e1 e2 ) = E ( e1 U е2 ) y ( е1 U е2 ) = = [ E ( e1 ) + E ( e2 ) ] ¥ ( e1 ~ € 2 ) E ( e1 ) y ( e1 U е2 ) + E ...
Page 1151
... disjoint closed subsets of R and if n is an integer , then there is an open set UCR such that AK CU and Ū □ B = 4. This is true since for each pɛ A ~ K2 there is an open set U ( p ) such that pɛ U ( p ) and U ( p ) ^ B = 4 ; by the ...
... disjoint closed subsets of R and if n is an integer , then there is an open set UCR such that AK CU and Ū □ B = 4. This is true since for each pɛ A ~ K2 there is an open set U ( p ) such that pɛ U ( p ) and U ( p ) ^ B = 4 ; by the ...
Page 1714
... disjoint . Let Ĉ1 and Ĉ2 be disjoint open sets containing C1 and C2 respectively . Then D1 = I - Ĉ1 and D2 = I - Ĉ2 are a pair of compact subsets whose union is I ; moreover , D1 CI1 and D2 C I1⁄2 . By Lemma 2.1 , there exists a pair of ...
... disjoint . Let Ĉ1 and Ĉ2 be disjoint open sets containing C1 and C2 respectively . Then D1 = I - Ĉ1 and D2 = I - Ĉ2 are a pair of compact subsets whose union is I ; moreover , D1 CI1 and D2 C I1⁄2 . By Lemma 2.1 , there exists a pair of ...
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BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense domain eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero