Linear Operators: Spectral theory |
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Page 958
... disjoint . Thus y ( e1 ) and y ( e ) are orthogonal whenever e1 and e , are disjoint . Hence if e1 and eg are disjoint then = = = e1 E ( e1 U е2 ) μ ( е1 U € 2 ) · [ E ( e1 ) + E ( e2 ) ] ¥ ( e1 U e2 ) E ( e1 ) y ( е1 U еg ) + E ( e2 ) ...
... disjoint . Thus y ( e1 ) and y ( e ) are orthogonal whenever e1 and e , are disjoint . Hence if e1 and eg are disjoint then = = = e1 E ( e1 U е2 ) μ ( е1 U € 2 ) · [ E ( e1 ) + E ( e2 ) ] ¥ ( e1 U e2 ) E ( e1 ) y ( е1 U еg ) + E ( e2 ) ...
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 U B 4. This is true since for each pe AK , ○ pɛ = there is an open set U ( p ) such that pe 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 U B 4. This is true since for each pe AK , ○ pɛ = there is an open set U ( p ) such that pe 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 I - Ĉ2 are a pair of compact subsets whose union is Ï ; moreover , D1 CI1 and D2 CI1⁄2 . By Lemma 2.1 , there exists a ...
... disjoint . Let Ĉ1 and Ĉ2 be disjoint open sets containing C1 and C2 respectively . Then D1 = I - Ĉ1 and D2 = I — Ĉ2 I - Ĉ2 are a pair of compact subsets whose union is Ï ; moreover , D1 CI1 and D2 CI1⁄2 . By Lemma 2.1 , there exists a ...
Contents
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 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 linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero