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 ) 00 Σ Ε ( δ ) α = i = 1 ∞ E ( U8 ; ) x , x ε H. i = 1 A spectral ...
... 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 ) 00 Σ Ε ( δ ) α = i = 1 ∞ E ( U8 ; ) x , x ε H. i = 1 A spectral ...
Page 894
... set functions whose values on a set σe are √g ( t ) E ( dt ) , E ( od ) , respectively . The integral Ss f ( s ) E ... Borel set 8 in S and every pair æ , æ * with x = X , x * € X * . It follows ( II.3.15 ) that E ( 8 ) = 0 . Thus if E ...
... set functions whose values on a set σe are √g ( t ) E ( dt ) , E ( od ) , respectively . The integral Ss f ( s ) E ... Borel set 8 in S and every pair æ , æ * with x = X , x * € X * . It follows ( II.3.15 ) that E ( 8 ) = 0 . Thus if E ...
Page 913
... Borel set e . Using the Lebesgue decomposition theorem ( III.4.14 ) , let { e } be a sequence of Borel sets such that = v ( en ) = 0 , and such that if e is a Borel subset of the complement en of e , and } v ; ( e ) ( e ) 0. Let σo be ...
... Borel set e . Using the Lebesgue decomposition theorem ( III.4.14 ) , let { e } be a sequence of Borel sets such that = v ( en ) = 0 , and such that if e is a Borel subset of the complement en of e , and } v ; ( e ) ( e ) 0. Let σo be ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero