## Linear Operators: Spectral theory |

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Page 861

Clearly if x - 1

+ y ) 2 ] = yz , ( To'y ) 2 = 7 ; ' ( yz ) , and if a = Tote , then az = To'z for every z e X.

Also Tea = e = T ; ' ( Tee ) = T ; ' ( ex ) = ( T e ) x = ax . Thus æ - 1

Clearly if x - 1

**exists**then Tr - Tc = T / T2-1 = 1 . If Til**exists**in B ( X ) , then Tx [ ( T2+ y ) 2 ] = yz , ( To'y ) 2 = 7 ; ' ( yz ) , and if a = Tote , then az = To'z for every z e X.

Also Tea = e = T ; ' ( Tee ) = T ; ' ( ex ) = ( T e ) x = ax . Thus æ - 1

**exists**and T ...Page 1057

Thus ( 2 ) gives 2 ( y ) F ( K * / ) ( u ) ( 21 ) -1/2 lim P S. X ; ( y ) { S efur f ( x − y ) dx

) day ER 2 ( y ) lim P Jgn lyn ZA ( Y ) e'wv dy } F ( ) ( u ) , provided only that the

limit in the braces in this last equation

Thus ( 2 ) gives 2 ( y ) F ( K * / ) ( u ) ( 21 ) -1/2 lim P S. X ; ( y ) { S efur f ( x − y ) dx

) day ER 2 ( y ) lim P Jgn lyn ZA ( Y ) e'wv dy } F ( ) ( u ) , provided only that the

limit in the braces in this last equation

**exists**. Thus , to complete the proof of the ...Page 1261

23 If an operator T has a closed linear extension there

linear extension T such that if T , is any closed linear extension of T then ICT . T is

called the closure of T. ( a ) There

closed ...

23 If an operator T has a closed linear extension there

**exists**a unique closedlinear extension T such that if T , is any closed linear extension of T then ICT . T is

called the closure of T. ( a ) There

**exists**a densely defined operator with noclosed ...

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