## Linear Operators: Spectral theory |

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Page 993

Then it

. , Qy is independent of V . Q . E . D . 16 THEOREM . If the bounded measurable

function o has its spectral set consisting of the single point m then , for some ...

Then it

**follows**from what has just been demonstrated that ay , = Qyuv , = Qy , i . e. , Qy is independent of V . Q . E . D . 16 THEOREM . If the bounded measurable

function o has its spectral set consisting of the single point m then , for some ...

Page 996

1 ( d ) it

seen that olf * ) Colp ) and from Lemma 12 ( c ) and the equation of = tf it

that olf * 9 ) contains no interior point of o ( 9 ) . Hence of * ) is a closed subset of

the ...

1 ( d ) it

**follows**from the above equation that f * 4 0 . From Lemma 12 ( b ) it isseen that olf * ) Colp ) and from Lemma 12 ( c ) and the equation of = tf it

**follows**that olf * 9 ) contains no interior point of o ( 9 ) . Hence of * ) is a closed subset of

the ...

Page 1486

If

6 . 35 , it

19 , T has no boundary values . Thus , by Definition 2 . 17 every linear ...

If

**follows**immediately that all the coefficients of t are bounded ; thus , by Theorem6 . 35 , it

**follows**that t has no boundary values at + 00 or at – 00 . By Theorem 2 .19 , T has no boundary values . Thus , by Definition 2 . 17 every linear ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive adjoint operator 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