## Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |

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Page 925

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

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

**sequence**of normal operators in H , all commuting with each other . Show that there exists a single Hermitian operator T such that each Nx is a Borel function of T. ( Hint : Use Theorem 2.1 and ...Page 959

Since Ueem = e , the

Since Ueem = e , the

**sequence**{ eembno m 2 1 } is an increasing**sequence**of sets whose union is ebn . Since Mo is countably additive on Boy Mo ( ebn ) = limm Mo ( eembn ) 2 k , and so for some m , Mo ( eem ) 2 Holeembn ) > k - .Page 1124

If E ,, E are in F and q ( En ) increases to the limit 9 ( E ) , then it follows from what we have already proved that E is an increasing

If E ,, E are in F and q ( En ) increases to the limit 9 ( E ) , then it follows from what we have already proved that E is an increasing

**sequence**of projections and E , SE . If Ec is the strong limit of En , then E. SE and Q ( E ) = Q ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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### Other editions - View all

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

### Common terms and phrases

additive adjoint operator algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem 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 vanishes vector zero