## Linear Operators: Spectral theory |

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Page 1223

How are we to choose its

the collection D , of all functions with one continuous derivative . If f and g are any

two such functions , we have ( iDf , g ) = So it ' ( ) g ( t ) dt = Steig ' ( + } dt + i ( ( 1 )

...

How are we to choose its

**domain**? A natural first guess is to choose as**domain**the collection D , of all functions with one continuous derivative . If f and g are any

two such functions , we have ( iDf , g ) = So it ' ( ) g ( t ) dt = Steig ' ( + } dt + i ( ( 1 )

...

Page 1248

The subspace M is called the initial

final

ranges of P * P and PP * are the initial and final

Proof .

The subspace M is called the initial

**domain**of P and PM ( = P $ ) is called thefinal

**domain**of P. 5 LEMMA . ... In this case PP * is also a projection and theranges of P * P and PP * are the initial and final

**domains**, respectively , of P.Proof .

Page 1249

Thus PP * is a projection whose range is N = PM , the final

complete the proof it will suffice to show that P * P is a projection if P is a partial

isometry . Let X , v EM , the initial

Pul2 ...

Thus PP * is a projection whose range is N = PM , the final

**domain**of P. Tocomplete the proof it will suffice to show that P * P is a projection if P is a partial

isometry . Let X , v EM , the initial

**domain**of P. Then the identity \ x + v2 = | Px +Pul2 ...

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