## Linear Operators: Spectral operators |

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Page 2236

Then , since T | Elen ) X is bounded ,

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

) ) 2 Dilf + g ) ( T ) ) n Dif ( T ) ) . This completes the proof of ( vi ) .

Then , since T | Elen ) X is bounded ,

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

) ) 2 Dilf + g ) ( T ) ) n Dif ( T ) ) . This completes the proof of ( vi ) .

**Statement**( iv ) ...Page 2239

Moreover ,

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

( fXe Xen ) x = lim T ( fxen ) x = T ( f ) x n 00 n00 100 by the operational calculus

for ...

Moreover ,

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

( fXe Xen ) x = lim T ( fxen ) x = T ( f ) x n 00 n00 100 by the operational calculus

for ...

Page 2476

We therefore see that , as asserted ,

that gi ( a ) = 0 for all a e e , . 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 e , belongs to

...

We therefore see that , as asserted ,

**statement**( 76 ) holds for each se H ' suchthat gi ( a ) = 0 for all a e e , . 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 e , belongs to

...

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### Contents

SPECTRAL OPERATORS | 1924 |

An Operational Calculus for Bounded Spectral | 1941 |

Part | 1950 |

Copyright | |

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