## Linear Operators: Spectral theory |

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Page 889

... of

measure E defined on the Borel sets in the plane and satisfying ( iv ) for every

Borel set d and ( v ) for every sequence { d ; } of

resolution ...

... of

**disjoint**Borel sets ( v ) SED , E ( 8 ) x = E ( US ; ) x , EH . i = 1 A spectralmeasure E defined on the Borel sets in the plane and satisfying ( iv ) for every

Borel set d and ( v ) for every sequence { d ; } of

**disjoint**Borel sets is called aresolution ...

Page 958

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. ylez ) and

that E ( e ) yley ) = 0 if e , and e , are

orthogonal whenever e , and e , are

yley u ...

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. ylez ) and

that E ( e ) yley ) = 0 if e , and e , are

**disjoint**. Thus y ( e , ) and ( ez ) areorthogonal whenever e , and e , are

**disjoint**. Hence if e , and eg are**disjoint**thenyley u ...

Page 1151

R = Un _ K . We observe that if A and B are

an integer , then there is an open set U CR such that An K , CU and U n B = $ .

This is true since for each pe An Kn there is an open set U ( p ) such that pe U ( p

) ...

R = Un _ K . We observe that if A and B are

**disjoint**closed subsets of R and if n isan integer , then there is an open set U CR such that An K , CU and U n B = $ .

This is true since for each pe An Kn there is an open set U ( p ) such that pe U ( p

) ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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additive Akad 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 Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero