Linear Operators: Spectral theory |
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Page 889
... set in the domain of a spectral measure satisfying ( iii ) is necessarily an ... Borel sets in the plane and which satisfies ( iv ) for every de B. This ... Borel sets ( v ) ∞ ∞ Σ E ( 8 ̧ ) x = E ( Ŭ 8 , ) x , i = 1 i = 1 x = $ . A ...
... set in the domain of a spectral measure satisfying ( iii ) is necessarily an ... Borel sets in the plane and which satisfies ( iv ) for every de B. This ... Borel sets ( v ) ∞ ∞ Σ E ( 8 ̧ ) x = E ( Ŭ 8 , ) x , i = 1 i = 1 x = $ . A ...
Page 894
... set functions whose values on a set oЄ are ( t ) E ( dt ) , E ( od ) , respectively . The integral ss f ( s ) E ( ds ... Borel set 8 in S and every pair æ , * with xe X , x * X * . It follows ( II.3.15 ) that E ( 8 ) = 0 . Thus if E and ...
... set functions whose values on a set oЄ are ( t ) E ( dt ) , E ( od ) , respectively . The integral ss f ( s ) E ( ds ... Borel set 8 in S and every pair æ , * with xe X , x * X * . It follows ( II.3.15 ) that E ( 8 ) = 0 . Thus if E and ...
Page 913
... Borel set e . Using the Lebesgue decomposition theorem ( III.4.14 ) , let { e } be a sequence of Borel sets such that Σv , ( en ) 0 , and such that if e is a Borel subset of the complement en of e , and Σ = v ( e ) ( e ) = 0. Let σ be ...
... Borel set e . Using the Lebesgue decomposition theorem ( III.4.14 ) , let { e } be a sequence of Borel sets such that Σv , ( en ) 0 , and such that if e is a Borel subset of the complement en of e , and Σ = v ( e ) ( e ) = 0. Let σ be ...
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