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

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Results 1-3 of 90

Page 1934

C = 0 , $ + 00 $ + it is

Corollary 11 . 3 . 14 , x ( 6 ) = 0 and thus x = ( $ I – T ' ) x ( $ ) = 0 . Q . E . D . → 4

THEOREM . Let T be a bounded spectral operator with resolution of the identity E

...

C = 0 , $ + 00 $ + it is

**seen**that x * x ( $ ) = 0 for all & and all x * e * * . Hence , byCorollary 11 . 3 . 14 , x ( 6 ) = 0 and thus x = ( $ I – T ' ) x ( $ ) = 0 . Q . E . D . → 4

THEOREM . Let T be a bounded spectral operator with resolution of the identity E

...

Page 2163

8 , it is

PCS ) – PLE E ( 0 : 0 ) | SK sup F ( $ i ) – F ( $ ) , where the supremum is taken

over those i and j for which 0 ; 0 ; is not void . If by the norm | | is understood the ...

8 , it is

**seen**that , for some constant K , ( iii ) = 1 Å PLEJE ( 6 ) - Ž PLEJEO ! ) | - Ž (PCS ) – PLE E ( 0 : 0 ) | SK sup F ( $ i ) – F ( $ ) , where the supremum is taken

over those i and j for which 0 ; 0 ; is not void . If by the norm | | is understood the ...

Page 2183

On the other hand , since PB contains all the elements of A ( B ) which have the

form ( ii ) , and since this set of elements is dense in A ( B ) , it will be

( T ) = A ( B ) . Moreover , it is clear that ( I – P ) B = R , and since R is closed , it ...

On the other hand , since PB contains all the elements of A ( B ) which have the

form ( ii ) , and since this set of elements is dense in A ( B ) , it will be

**seen**that PA( T ) = A ( B ) . Moreover , it is clear that ( I – P ) B = R , and since R is closed , it ...

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