## Linear Operators: Spectral theory |

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Page 925

17 For operators A , B , and C in Hilbert space show that ( a ) ASB and BSA

A = B ; ( b ) ASB and BSC

SB

17 For operators A , B , and C in Hilbert space show that ( a ) ASB and BSA

**imply**A = B ; ( b ) ASB and BSC

**imply**AS C ; ( c ) A , SA , and SB ... 18 Show that 0 SASB

**implies**that A ' < B2 if A and B commute , but not if A and B fail to commute .Page 1124

Hence Exn ) 2 = \ E , « 12 for each n , so that Exn = E , Xn and E = E . That is , q (

E ) = Q ( E )

F and 9 ( En ) increases to the limit 9 ( E ) , then it follows from what we have ...

Hence Exn ) 2 = \ E , « 12 for each n , so that Exn = E , Xn and E = E . That is , q (

E ) = Q ( E )

**implies**E = E . Similarly , Q ( E ) = Q ( E )**implies**E s E . If En , E are inF and 9 ( En ) increases to the limit 9 ( E ) , then it follows from what we have ...

Page 1229

... ( e + te _ ) ) - ( ( + + d _ ) , T * ( e7 + e _ ) ) } = ~ i { ( id - id _ ) , ( ette _ ) ) - ( ( d +

+ d _ ) , ( ie - - ie _ ) ) } = ~ i { 2i ( d , ( + ) - 2i ( d _ , e _ ) } = 2 { ( dt , ( + ) - ( d _ , e _

) } = 0 , i . e . , if and only if ( dt , ex ) = ( d _ , e _ ) . This clearly

...

... ( e + te _ ) ) - ( ( + + d _ ) , T * ( e7 + e _ ) ) } = ~ i { ( id - id _ ) , ( ette _ ) ) - ( ( d +

+ d _ ) , ( ie - - ie _ ) ) } = ~ i { 2i ( d , ( + ) - 2i ( d _ , e _ ) } = 2 { ( dt , ( + ) - ( d _ , e _

) } = 0 , i . e . , if and only if ( dt , ex ) = ( d _ , e _ ) . This clearly

**implies**that 10 _ 12...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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