## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 2236

Then , since T | E ( en ) X is bounded ,

calculus of bounded operators ( cf . ... ( g ( T ) ) , it is apparent that D ( f ( T ) + g ( T

) ) 2 D ( If + g ) ( T ) ) D ( f ( T ) ) . This completes the proof of ( vi ) .

...

Then , since T | E ( en ) X is bounded ,

**statements**( i ) and ( ii ) and the functionalcalculus of bounded operators ( cf . ... ( g ( T ) ) , it is apparent that D ( f ( T ) + g ( T

) ) 2 D ( If + g ) ( T ) ) D ( f ( T ) ) . This completes the proof of ( vi ) .

**Statement**( iv )...

Page 2239

Moreover ,

Letting e e E , and x e E ( e ) X , we have T ' ( fxe ) x = lim T ' ( fxe ) E ( en ) x = lim

T ( fXe XenJx 000 = lim T ( fxen ) . = T ( f ) . n + 00 by the operational calculus for ...

Moreover ,

**statement**( g ) follows from Corollary 7 .**Statement**( d ) is obvious .Letting e e E , and x e E ( e ) X , we have T ' ( fxe ) x = lim T ' ( fxe ) E ( en ) x = lim

T ( fXe XenJx 000 = lim T ( fxen ) . = T ( f ) . n + 00 by the operational calculus for ...

Page 2476

We therefore see that , as asserted ,

that gi ( a ) = 0 for all a key . Since ( H + V , H ) is a closed subspace of H ' ( cf .

Lemma 2 ) it follows that every s e H ' such that gi ( a ) = 0 for all a € e , belongs to

...

We therefore see that , as asserted ,

**statement**( 76 ) holds for each se H ' suchthat gi ( a ) = 0 for all a key . Since ( H + V , H ) is a closed subspace of H ' ( cf .

Lemma 2 ) it follows that every s e H ' such that gi ( a ) = 0 for all a € e , belongs to

...

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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