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。• · Мо To see that μo is countably additive on Bo let e ,, n ≥ 1 , be disjoint sets in Bo whose union e is also in B 。. Let rn = en en + 10 . so that E ( r ) g → 0 for every g in L2 ( R ) and , by Lemma 5 , 0 ( g , y ...
... additive on B。• · Мо To see that μo is countably additive on Bo let e ,, n ≥ 1 , be disjoint sets in Bo whose union e is also in B 。. Let rn = en en + 10 . so that E ( r ) g → 0 for every g in L2 ( R ) and , by Lemma 5 , 0 ( g , y ...
Page 959
... countably additive on Bo , Mo ( eb2 ) = limm Mo ( eembn ) ≥k , and so for some m , μo ( eem ) ≥ μo ( еembn ) > k − ɛ . This shows that the set function μ is uniquely defined on B. мо Next we show that u is countably additive . For ...
... countably additive on Bo , Mo ( eb2 ) = limm Mo ( eembn ) ≥k , and so for some m , μo ( eem ) ≥ μo ( еembn ) > k − ɛ . This shows that the set function μ is uniquely defined on B. мо Next we show that u is countably additive . For ...
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