## Linear Operators: Spectral theory |

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Page 876

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

( 1 + N ) < ly + Niel2 = | ( y + Nie ) ( y + Nie ) * ) = ( y + Nie ) ( y - Nie ) \ y2 + N el = \

y2I + N2 . Since this inequality must hold for all real N , a contradiction is ...

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

**hence**11 + N ly + Niel .**Hence**( 1 + N ) < ly + Niel2 = | ( y + Nie ) ( y + Nie ) * ) = ( y + Nie ) ( y - Nie ) \ y2 + N el = \

y2I + N2 . Since this inequality must hold for all real N , a contradiction is ...

Page 1027

scalar 2 belongs to the spectrum of ET . Then , for some non - zero æ in EH , we

have ETx = 2x . Then Tx = x + 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 2 belongs to the spectrum of ET . Then , for some non - zero æ in EH , we

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

...

Page 1227

and D are clearly linear subspaces of D ( T * ) , 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 æ e D.**Hence**D is closed . Similarly , D is closed . Since D4and D are clearly linear subspaces of D ( T * ) , 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

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Copyright | |

57 other sections not shown

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