Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 876
... hence 1+ N ≤ y + Nie . Hence ( 1 + N ) 2 ≤ y + Nie | 2 = | ( y + Nie ) ( y + Nie ) * | = | ( y + Nie ) ( y - Nie ) | | ( y + Nie ) ( y + Nie ) * | \ y2 + N2e \ ≤ \ y2 \ + N2 . Since this inequality must hold for all real N , a ...
... hence 1+ N ≤ y + Nie . Hence ( 1 + N ) 2 ≤ y + Nie | 2 = | ( y + Nie ) ( y + Nie ) * | = | ( y + Nie ) ( y - Nie ) | | ( y + Nie ) ( y + Nie ) * | \ y2 + N2e \ ≤ \ y2 \ + N2 . Since this inequality must hold for all real N , a ...
Page 1027
... hence for some non - zero x in § we have Tx λα , 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 belongs to the spectrum of ET . Then , for ...
... hence for some non - zero x in § we have Tx λα , 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 belongs to the spectrum of ET . Then , for ...
Page 1227
... Hence T * x = ix , or x = D. Hence D is closed . Similarly , xe 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 ...
... Hence T * x = ix , or x = D. Hence D is closed . Similarly , xe 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 ...
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