## Linear Operators: Spectral operators |

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Page 2152

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. To

C ) , let be a closed subset of the complex plane and let M ( 8 ) = { x | x € X , g ( x )

$ 8 } . It will be shown that M ( S ) is closed . For every & we have g ( x ) = o ( T ) ...

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. To

**prove**(C ) , let be a closed subset of the complex plane and let M ( 8 ) = { x | x € X , g ( x )

$ 8 } . It will be shown that M ( S ) is closed . For every & we have g ( x ) = o ( T ) ...

Page 2236

This

+ g ( T ) ) and let { en } be as above . Then , since T | Elen ) X is bounded ,

statements ( i ) and ( ii ) and the functional calculus of bounded operators ( cf. VII.

3.10 ) ...

This

**proves**( iii ) . The proof of ( viii ) is evident . To**prove**( vi ) , let x be in Dif ( T )+ g ( T ) ) and let { en } be as above . Then , since T | Elen ) X is bounded ,

statements ( i ) and ( ii ) and the functional calculus of bounded operators ( cf. VII.

3.10 ) ...

Page 2459

If xn € { ac ( H ) and lim , -a Xn = x , then , by what we have already

may write x = yı + y2 + 4s , where yı e Lac ( H ) and Y2 , Yz are orthogonal to Eac

( H ) . ... Using this last fact , it is easy to

.

If xn € { ac ( H ) and lim , -a Xn = x , then , by what we have already

**proved**, wemay write x = yı + y2 + 4s , where yı e Lac ( H ) and Y2 , Yz are orthogonal to Eac

( H ) . ... Using this last fact , it is easy to

**prove**assertion ( c ) of the present lemma.

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### Contents

SPECTRAL OPERATORS 1937 1941 1945 XV Spectral Operators | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

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