## 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 N . 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 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 ebr . Since Mo is countably additive on Bo , Molebn ) = limon

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

...

Since Ueem = e , the

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

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

...

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

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero