Linear Operators: Spectral theory |
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Page 897
... countably additive in the strong operator topology . PROOF . If { 8 } is a sequence of Borel sets decreasing to the ... countably additive self adjoint spectral measure " without ambiguity . 4 COROLLARY . A bounded normal operator T ...
... countably additive in the strong operator topology . PROOF . If { 8 } is a sequence of Borel sets decreasing to the ... countably additive self adjoint spectral measure " without ambiguity . 4 COROLLARY . A bounded normal operator T ...
Page 958
... additive on B 。. μo - To see that is countably additive on Bo let e , n ≥ 1 , be disjoint sets in Bo whose union e is also in Bo . Let r1 = en Uen + 1 ~ . . . , so that E ( r ) g → 0 for every g in L2 ( R ) and , by Lemma 5 , 0 ( g ...
... additive on B 。. μo - To see that is countably additive on Bo let e , n ≥ 1 , be disjoint sets in Bo whose union e is also in Bo . Let r1 = en Uen + 1 ~ . . . , so that E ( r ) g → 0 for every g in L2 ( R ) and , by Lemma 5 , 0 ( g ...
Page 959
... countably additive Мо on Bo , Mo ( ebn ) limm Mo ( eembn ) k , and so for some m , uo ( eem ) Mo ( eembn ) > k - ɛ . This shows that the set function u is uniquely defined on B. = Next we show that u is countably additive . For this ...
... countably additive Мо on Bo , Mo ( ebn ) limm Mo ( eembn ) k , and so for some m , uo ( eem ) Mo ( eembn ) > k - ɛ . This shows that the set function u is uniquely defined on B. = Next we show that u is countably additive . For this ...
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