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

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Page 1986

Hence , to complete the proof of the theorem it suffices to

formula ( 5 ) . ... ( u ) du } dt = lim c { 1 , - oglu ) duzat = lim - 1 po a 07T1 - 00 sin

als – u ) $ - U - - P ( u ) du 1 poo sin at and so equation ( 5 ) may be

by.

Hence , to complete the proof of the theorem it suffices to

**establish**the inversionformula ( 5 ) . ... ( u ) du } dt = lim c { 1 , - oglu ) duzat = lim - 1 po a 07T1 - 00 sin

als – u ) $ - U - - P ( u ) du 1 poo sin at and so equation ( 5 ) may be

**established**by.

Page 2212

We next

fz Xocp ) so that , using ( vii ) , we have g ( a ) = 0 for 1€ 07 . 0 ( F ) = 077 Also T ...

Thus if fz ( a ) = fw ( a ) on 0 , = 0w , equation ( ix ) will be

...

We next

**establish**the equation ( viii ) fpz = f Xo ( F ) , Fe B . To prove this , let g =fz Xocp ) so that , using ( vii ) , we have g ( a ) = 0 for 1€ 07 . 0 ( F ) = 077 Also T ...

Thus if fz ( a ) = fw ( a ) on 0 , = 0w , equation ( ix ) will be

**established**. Hence we...

Page 2234

Let x be in E ( e ) X and let x be in D ( f ( T | E ( e ) x ) ) . Then by Definition 8 ,

since ( ii ) has already been

contained in U , f ( T | E ( e ) X ) x = lim f ( T | F ( en ) E ( e ) x ) F ( en ) x no = lim f (

T | E ...

Let x be in E ( e ) X and let x be in D ( f ( T | E ( e ) x ) ) . Then by Definition 8 ,

since ( ii ) has already been

**established**for bounded Borel sets with closurescontained in U , f ( T | E ( e ) X ) x = lim f ( T | F ( en ) E ( e ) x ) F ( en ) x no = lim f (

T | E ...

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