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 e are disjoint then ( ees ) = = - = E ( e , e2 ) ( e1e2 ) [ E ( e1 ) + E ( e2 ) ] y ( e1 ~ € 2 ) E ( e ) y ( e1 e2 ) + E ( e2 ) y ...
... disjoint . Thus y ( e1 ) and y ( e ) are orthogonal whenever e1 and e , are disjoint . Hence if e1 and e are disjoint then ( ees ) = = - = E ( e , e2 ) ( e1e2 ) [ E ( e1 ) + E ( e2 ) ] y ( e1 ~ € 2 ) E ( e ) y ( e1 e2 ) + E ( e2 ) y ...
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 n = there is an open set U ( p ) such that p € U ( p ) and U ( p ) ^ B $ ; 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 n = there is an open set U ( p ) such that p € U ( p ) and U ( p ) ^ B $ ; by the ...
Page 1187
... disjoint and their union is the whole plane . The resolvent set p ( T ) is open and the resolvent R ( λ ; T ) is an analytic function of 2 and satisfies the resolvent equation ww R ( λ ; T ) — R ( μ ; T ) = ( u - 2 ) R ( 2 ; T ) R ( μ ...
... disjoint and their union is the whole plane . The resolvent set p ( T ) is open and the resolvent R ( λ ; T ) is an analytic function of 2 and satisfies the resolvent equation ww R ( λ ; T ) — R ( μ ; T ) = ( u - 2 ) R ( 2 ; T ) R ( μ ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero