Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 889
... Borel sets ( v ) Σ E ( 8 , ) x i = 1 = ∞ E ( US ) x , x = H. i = 1 A spectral measure E defined on the Borel sets in the plane and satis- fying ( iv ) for every Borel set 8 and ( v ) for every sequence { 8 } of dis- joint Borel sets is ...
... Borel sets ( v ) Σ E ( 8 , ) x i = 1 = ∞ E ( US ) x , x = H. i = 1 A spectral measure E defined on the Borel sets in the plane and satis- fying ( iv ) for every Borel set 8 and ( v ) for every sequence { 8 } of dis- joint Borel sets is ...
Page 913
... Borel set e . Using the Lebesgue decomposition theorem ( III.4.14 ) , let { e } be a sequence of Borel sets such that Σv , ( e , ) = 0 , and such that if e is a Borel subset of the complement en of e , and Σ = v ( e ) 2-1 ( e ) = 0. Let ...
... Borel set e . Using the Lebesgue decomposition theorem ( III.4.14 ) , let { e } be a sequence of Borel sets such that Σv , ( e , ) = 0 , and such that if e is a Borel subset of the complement en of e , and Σ = v ( e ) 2-1 ( e ) = 0. Let ...
Page 1900
... Borel function , X.1 ( 891 ) Borel measurable function , X.I ( 891 ) Borel measure ( or Borel - Lebesgue measure ) , construction of , ( 139 ) , III.13.8 ( 223 ) Borel - Stieltjes measure , ( 142 ) Bound , of an operator , II.3.5 ( 60 ) ...
... Borel function , X.1 ( 891 ) Borel measurable function , X.I ( 891 ) Borel measure ( or Borel - Lebesgue measure ) , construction of , ( 139 ) , III.13.8 ( 223 ) Borel - Stieltjes measure , ( 142 ) Bound , of an operator , II.3.5 ( 60 ) ...
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 Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm 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 transform unique unitary vanishes vector zero