## Linear Operators: Spectral theory |

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Page 1064

But it is

, if f is in Co ( I ) and vanishes outside a bounded set . Next let | be in L , ( E " ) ,

and let { m } be a sequence of functions in ( ° ( E ) , each vanishing outside a ...

But it is

**clear**that gm ( x ) + g ( x ) for all z . Hence q = g , proving that gl , $ 1,1 \ | \, if f is in Co ( I ) and vanishes outside a bounded set . Next let | be in L , ( E " ) ,

and let { m } be a sequence of functions in ( ° ( E ) , each vanishing outside a ...

Page 1652

Then , since | F 2 | F | , for each F in H ( * ) ( I ) , it is

some F in Lề ( 1 ) . Similarly , since Fl ( ) 210F \ , for each F in H ( * ) ( I ) and each

index J such that Jsk , it is

Then , since | F 2 | F | , for each F in H ( * ) ( I ) , it is

**clear**that { Fn } converges tosome F in Lề ( 1 ) . Similarly , since Fl ( ) 210F \ , for each F in H ( * ) ( I ) and each

index J such that Jsk , it is

**clear**that if J Sk , the sequence { adF , } converges to ...Page 1725

Since the function qe defined by qe ( x ) = 9 ( ( 1- € ) x ) converges to q in the

norm of CP ( I ) as ε = 0 by Lemma 2.5 , it is

loss of generality that q is in C % ( I ) . This will be assumed in what follows . Let K

be a ...

Since the function qe defined by qe ( x ) = 9 ( ( 1- € ) x ) converges to q in the

norm of CP ( I ) as ε = 0 by Lemma 2.5 , it is

**clear**that we may assume withoutloss of generality that q is in C % ( I ) . This will be assumed in what follows . Let K

be a ...

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