Linear Operators, Part 2 |
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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 dЄ B. This ... Borel sets Ε ( v ) ∞ ∞ Σ Ε ( δ ) α = E ( Ŭ 8 , ) x , x = H. 2 = 1 i = 1 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 dЄ B. This ... Borel sets Ε ( v ) ∞ ∞ Σ Ε ( δ ) α = E ( Ŭ 8 , ) x , x = H. 2 = 1 i = 1 A ...
Page 894
... set functions whose values on a set σe g ( t ) E ( de ) . are E ( od ) , ก respectively . The integral ss f ( s ) E ... Borel set 8 in S and every pair x , x * with x = X , x * = X * . It follows ( II.3.15 ) that E ( 8 ) Thus if E and A ...
... set functions whose values on a set σe g ( t ) E ( de ) . are E ( od ) , ก respectively . The integral ss f ( s ) E ... Borel set 8 in S and every pair x , x * with x = X , x * = X * . It follows ( II.3.15 ) that E ( 8 ) Thus if E and A ...
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 Σv , ( e ) v , ( e ) = 0. Let o ...
... 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 Σv , ( e ) v , ( e ) = 0. Let o ...
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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 domain 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 linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma 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