## 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 | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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