## Linear Operators: Spectral operators |

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Page 1986

We next

F is one - to - one on 0 and that o ( s ) = ( F20 ) ( - s ) , which proves that F = 0 and

thus that the inverse ( F - 10 ) ( s ) = ( F20 ) ( - s ) is everywhere defined and ...

We next

**establish**the inversion formula ( 5 ) . Once this is done , it will follow thatF is one - to - one on 0 and that o ( s ) = ( F20 ) ( - s ) , which proves that F = 0 and

thus that the inverse ( F - 10 ) ( s ) = ( F20 ) ( - s ) is everywhere defined and ...

Page 2101

Such a formula has been

) the Boolean algebra generated by E , and E , is bounded , ( ii ) X is weakly

complete , and ( iii ) the E - measure of the boundary of o is zero . Further work

along ...

Such a formula has been

**established**by Foguel [ 1 ] under the hypotheses that ( i) the Boolean algebra generated by E , and E , is bounded , ( ii ) X is weakly

complete , and ( iii ) the E - measure of the boundary of o is zero . Further work

along ...

Page 2212

We next

fa Xocp ) so that , using ( vii ) , we have g ( X ) = 0 for 1€ 0 , 0 ( F ) = Ofz . Also T ( g

) Fx = T ... Thus if fz ( a ) = fw ( a ) on 0 , = Ow , equation ( ix ) will be

We next

**establish**the equation ( viii ) ffo = fo Xo ( F ) Fe B . To prove this , let g =fa Xocp ) so that , using ( vii ) , we have g ( X ) = 0 for 1€ 0 , 0 ( F ) = Ofz . Also T ( g

) Fx = T ... Thus if fz ( a ) = fw ( a ) on 0 , = Ow , equation ( ix ) will be

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