## Linear Operators: Spectral theory |

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Page 1223

Consider , as an example , an operator which will be studied in greater detail in

the next chapter : the differential operator iD = i ( d / dt ) in the space L2 ( 0 , 1 ) .

How are we to choose its

Consider , as an example , an operator which will be studied in greater detail in

the next chapter : the differential operator iD = i ( d / dt ) in the space L2 ( 0 , 1 ) .

How are we to choose its

**domain**? A natural first guess is to choose as**domain**...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 ( = PH ) 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 E M , the initial

Pola ...

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 E M , the initial

**domain**of P . Then the identity ( x + 0 / 2 = \ Px +Pola ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

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