Linear Operators: Spectral theory |
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Page 876
... hence and │1 + N ≤ y + Nie . Hence ( 1 + N ) 2 ≤ \ y + Nie | 2 = | ( 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 ...
... hence and │1 + N ≤ y + Nie . Hence ( 1 + N ) 2 ≤ \ y + Nie | 2 = | ( 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 ...
Page 1027
... hence for some non - zero x in § we have Tx = λα , and hence , since T TE , we have ( ET ) ( Ex ) = λEx . Hence 2 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 = λα , and hence , since T TE , we have ( ET ) ( Ex ) = λEx . Hence 2 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 xe 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 ) , D4 , and D_ are mutually orthogonal , and that their sum ...
... Hence T * x = ix , or xe 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 ) , D4 , and D_ are mutually orthogonal , and that their sum ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero