## Linear Operators: Spectral theory |

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Page 876

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

= \ y2 + Nael \ y2I + N2 . Since this inequality must hold for all real N , a

contradiction is ...

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

**hence**11 + N Sly + Niel .**Hence**( 1 + N ) 2 < \ y + Nie | 2 = | ( y + Nie ) ( y + Nie ) * 1 = | ( y + Nie ) ( y - Nie )= \ y2 + Nael \ y2I + N2 . Since this inequality must hold for all real N , a

contradiction is ...

Page 1027

5 shows that a is an eigenvalue and

Tx = ax , and

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

5 shows that a is an eigenvalue and

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

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

Page 1227

and D _ are clearly linear subspaces of D ( 7 ' * ) , it remains to show that the

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

**Hence**T * x = ix , or x € Dt .**Hence**Dt is closed . Similarly , D is closed . Since Dand D _ are clearly linear subspaces of D ( 7 ' * ) , it remains to show that the

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

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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