## Linear Operators, Part 2 |

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Page 876

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

\ y2 + Nael = \ y2 ! + 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 + Niel2 = | ( y + Nie ) ( y + Nie ) * 1 = | ( y + Nie ) ( y - Nie ) =\ y2 + Nael = \ y2 ! + N2 . Since this inequality must hold for all real N , a

contradiction ...

Page 1027

scalar , belongs to the spectrum of ET . Then , for some non - zero X in EH , we

have ETx = 2x . Then Tx = 2x + y , where y belongs to the subspace ( I - E ) H ,

and ...

**Hence**a belongs to the spectrum of ET . Conversely , suppose that a non - zeroscalar , belongs to the spectrum of ET . Then , for some non - zero X in EH , we

have ETx = 2x . Then Tx = 2x + y , where y belongs to the subspace ( I - E ) H ,

and ...

Page 1227

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

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

.

**Hence**T * x = ix , or x e Dr .**Hence**D . is closed . Similarly , D _ is closed . SinceDt and D _ are clearly linear subspaces of D ( T * ) , it remains to show that the

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

.

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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