Linear Operators: Spectral theory |
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Page 876
... hence 1+ N ≤y + Nie . Hence = ( 1 + N ) 2 ≤ \ y + Nie2 = | ( y + Nie ) ( y + Nie ) * | = | ( y + Nie ) ( y − Nie ) | \ y2 + N2e ] ≤ \ y2 + N2 . Since this inequality must hold for all real N , a contradiction is obtained by placing ...
... hence 1+ N ≤y + Nie . Hence = ( 1 + N ) 2 ≤ \ y + Nie2 = | ( y + Nie ) ( y + Nie ) * | = | ( y + Nie ) ( y − Nie ) | \ y2 + N2e ] ≤ \ y2 + N2 . Since this inequality must hold for all real N , a contradiction is obtained by placing ...
Page 1027
... hence for some non - zero x in § we have Tx λπ , and hence , since T TE , we have ( ET ) ( Ex ) = λEx . Hence λ belongs to the spectrum of ET . Conversely , suppose that a non - zero scalar λ belongs to the spectrum of ET . Then , for ...
... hence for some non - zero x in § we have Tx λπ , and hence , since T TE , we have ( ET ) ( Ex ) = λEx . Hence λ belongs to the spectrum of ET . Conversely , suppose that a non - zero scalar λ belongs to the spectrum of ET . Then , for ...
Page 1227
... Hence T * x ix , or x = D. Hence D is closed . Similarly , D is closed . Since D , and D are clearly linear subspaces of D ( T * ) , it remains to show that the spaces D ( T ) , D. , and D are mutually orthogonal , and that their sum is ...
... Hence T * x ix , or x = D. Hence D is closed . Similarly , D is closed . Since D , and D are clearly linear subspaces of D ( T * ) , it remains to show that the spaces D ( T ) , D. , and D are mutually orthogonal , and that their sum is ...
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