## Linear Operators: Self Adjoint Operators in Hilbert Space. Spectral theory. Part II |

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Page 925

B ; 16 Let N ,, 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 ) .

B ; 16 Let N ,, 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

Since Ueem = e , the

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

Mo ( eembn ) 2k , and so for some m , Moleem ) Moleembn ) > k - ε . This shows ...

Since Ueem = e , the

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

Mo ( eembn ) 2k , and so for some m , Moleem ) Moleembn ) > k - ε . This shows ...

Page 1124

That is , Q ( E ) = Q ( E ) implies E = E. Similarly , y ( E ) SQ ( E ) implies ESE . If En

, E are in F and q ( En ) increases to the limit ( 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 , y ( E ) SQ ( E ) implies ESE . If En

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

have already proved that En is an increasing

**sequence**of projections and En > E.

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

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