## Linear Operators: Spectral theory |

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Page 925

16 Let N1 , N , , . . . 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 N1 , N , , . . . 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

Since Ueem = e , the

sets whose union is ebn . Since Mo is countably additive on Bo , Mo ( ebn ) =

limm Moleembn ) 2 k , and so for some m , Moleem ) 2 Holeembn ) > k - ε . This

shows ...

Since Ueem = e , the

**sequence**{ eembn , m 2 1 } is an increasing**sequence**ofsets whose union is ebn . Since Mo is countably additive on Bo , Mo ( ebn ) =

limm Moleembn ) 2 k , and so for some m , Moleem ) 2 Holeembn ) > k - ε . This

shows ...

Page 1124

That is , Q ( E ) = Q ( E ) implies E = E . Similarly , 4 ( E ) = Q ( E ) implies E s E . If

En , E are in F and q ( 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 = E . Similarly , 4 ( E ) = Q ( E ) implies E s E . If

En , E are in F and q ( 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 E ...### What people are saying - Write a review

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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