## Linear Operators: Spectral theory |

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Page 925

16 Let N , , N , , . . . be a countable

commuting with each other . Show that there exists a single Hermitian operator T

such that each Ne is a Borel function of T . ( Hint : Use Theorem 2 . 1 and

Exercise 15 ) .

16 Let N , , N , , . . . be a countable

**sequence**of normal operators in ý , allcommuting with each other . Show that there exists a single Hermitian operator T

such that each Ne is a Borel function of T . ( Hint : Use Theorem 2 . 1 and

Exercise 15 ) .

Page 959

prove the uniqueness of the limit it will suffice to show that if Molebn ) 2 k for

some n , then , for every e > 0 , Moleem ) > k - e for some m . Since Ueem = e , the

ebn .

prove the uniqueness of the limit it will suffice to show that if Molebn ) 2 k for

some n , then , for every e > 0 , Moleem ) > k - e for some m . Since Ueem = e , the

**sequence**{ eembn , m 2 1 } is an increasing**sequence**of sets whose union isebn .

Page 1124

That is , Q ( E ) = Q ( E ) implies E = Eq . Similarly , q ( E ) < 9 ( E ) implies E s Eq .

If En , E are in F and 9 ( En ) increases to the limit 9 ( E ) , then it follows from what

we have already proved that En is an increasing

That is , Q ( E ) = Q ( E ) implies E = Eq . Similarly , q ( E ) < 9 ( E ) implies E s Eq .

If En , E are in F and 9 ( En ) increases to the limit 9 ( E ) , then it follows from what

we have already proved that En is an increasing

**sequence**of projections and ...### What people are saying - Write a review

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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