## Linear Operators, Part 2 |

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Page 861

If To

Tote , then az = Totz for every ze X . Also xa = Tya = e = T ; ' ( Tee ) = T ; " ex ) = ( T

' e ) x = ax . Thus x - 1

...

If To

**exists**in B ( x ) , then Tx [ ( T2 + y ) 2 ] = yz , ( To ' y ) 2 = T ( yz ) , and if a =Tote , then az = Totz for every ze X . Also xa = Tya = e = T ; ' ( Tee ) = T ; " ex ) = ( T

' e ) x = ax . Thus x - 1

**exists**and T ; % = x - 1 % . 2 DEFINITION . An element « in...

Page 1057

Thus , to complete the proof of the present lemma , it suffices to show that [ 2 ( y )

12 ( y ) ( 3 ) O ( u ) = P eiyu dy = lim eiyu dy E - > JeslySR y " Jen lyn R - 00

for each u . By Lemma 2 , the integral 0 ( tu )

Thus , to complete the proof of the present lemma , it suffices to show that [ 2 ( y )

12 ( y ) ( 3 ) O ( u ) = P eiyu dy = lim eiyu dy E - > JeslySR y " Jen lyn R - 00

**exists**for each u . By Lemma 2 , the integral 0 ( tu )

**exists**if 0 ( u )**exists**and t > 0 ; and ...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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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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