## Linear Operators: Spectral theory |

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Page 925

16 Let Ni , N2 , . . . be a countable

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

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

Exercise 15 ) .

16 Let Ni , N2 , . . . be a countable

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

such that each N . 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 Holebn ) 2 k for some

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

ebm .

prove the uniqueness of the limit it will suffice to show that if Holebn ) 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 isebm .

Page 1124

Hence , if we choose a countable set { E ; } CF such that { 9 ( E ; ) } is dense in the

range of the function q , then each E in F is the limit either of an increasing or of a

decreasing

Hence , if we choose a countable set { E ; } CF such that { 9 ( E ; ) } is dense in the

range of the function q , then each E in F is the limit either of an increasing or of a

decreasing

**sequence**of projections in F . We shall show below that there ...### What people are saying - Write a review

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

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