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 e , and e are disjoint then ¥ ( e1 ○ e2 ) = E ( ej ○ е2 ) ¥ ( е1 ~ еg ) = = = [ E ( e1 ) + E ( e2 ) ] ¥ ( e1 U eg ) E ( e1 ) ¥ ( e1 ~ e2 ) ...
... disjoint . Thus y ( e1 ) and y ( e ) are orthogonal whenever e1 and e , are disjoint . Hence if e , and e are disjoint then ¥ ( e1 ○ e2 ) = E ( ej ○ е2 ) ¥ ( е1 ~ еg ) = = = [ E ( e1 ) + E ( e2 ) ] ¥ ( e1 U eg ) E ( e1 ) ¥ ( e1 ~ e2 ) ...
Page 959
... disjoint sequence in B. It is clear that μ ( Uan ) ≥μ ( an ) , so that , if μ ( a ) = ∞ for any n , the equation μ ( Ua , ) = Σμ ( an ) is trivially true . Hence we may and shall assume that μ ( a ) < ∞ for each n . Consequently ...
... disjoint sequence in B. It is clear that μ ( Uan ) ≥μ ( an ) , so that , if μ ( a ) = ∞ for any n , the equation μ ( Ua , ) = Σμ ( an ) is trivially true . Hence we may and shall assume that μ ( a ) < ∞ for each n . Consequently ...
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 pЄ A K there is an open set U ( p ) such that p € U ( p ) and U ( p ) ^ B = 6 ; 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 pЄ A K there is an open set U ( p ) such that p € U ( p ) and U ( p ) ^ B = 6 ; by the ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Copyright | |
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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 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 real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero