## Linear Operators: Spectral theory |

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Page 876

Then ( y + Nie ) ( 2 ) = y ( 2 ) + Ni = i ( 1 + N ) , and

= \ y2 + Nael = \ 21 + N2 . Since this inequality must hold for all real N , a

contradiction ...

Then ( y + Nie ) ( 2 ) = y ( 2 ) + Ni = i ( 1 + N ) , and

**hence**11 + NI Sly + Niel .**Hence**( 1 + N ) 2 5 \ y + Nie | 2 = | ( y + Nie ) ( y + Nie ) * 1 = | ( y + Nie ) ( y - Nie )= \ y2 + Nael = \ 21 + N2 . Since this inequality must hold for all real N , a

contradiction ...

Page 1027

5 shows that 2 is an eigenvalue and

Tx = kx , and

to the spectrum of ET . Conversely , suppose that a non - zero scalar belongs to ...

5 shows that 2 is an eigenvalue and

**hence**for some non - zero æ in H we haveTx = kx , and

**hence**, since T = TE , we have ( ET ) ( Ex ) = 1Ex .**Hence**a belongsto the spectrum of ET . Conversely , suppose that a non - zero scalar belongs to ...

Page 1227

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 is D ( T * ) .

**Hence**T * x = ix , or xe Dr .**Hence**Dc is closed . Similarly , D is closed . Since Dand 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 is D ( T * ) .

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero