## Linear Operators: Spectral theory |

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Page 876

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

\ y2 + Nel = \ y ? I + N2 . Since this inequality must hold for all real N , a

contradiction ...

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

**hence**11 + N = ly + Niel .**Hence**( 1 + N ) 2 = ly + Nie | 2 = [ ( y + Nie ) ( y + Nie ) * ) = | ( y + Nie ) ( y - Nie ) =\ y2 + Nel = \ y ? I + N2 . Since this inequality must hold for all real N , a

contradiction ...

Page 1027

5 shows that 2 is an eigenvalue and

= lx , and

the spectrum of ET . Conversely , suppose that a non - zero scalar 1 belongs to ...

5 shows that 2 is an eigenvalue and

**hence**for some non - zero x in H we have Tx= lx , and

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

Page 1227

. and D _ are clearly linear subspaces of D ( T * ) , it remains to show that the

spaces D ( T ) , Dt , and D are mutually orthogonal , and that their sum is D ( T * ) .

**Hence**T * x = ix , or x e Dt .**Hence**Dt 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 ) , Dt , and D are mutually orthogonal , and that their sum is D ( T * ) .

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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