## 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 Qy , = 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 * 9 ) Colp ) and from Lemma 12 ( c ) and the equation of = tf it

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

1 ( d ) it

**follows**from the above equation that f * 9 # 0 . From Lemma 12 ( b ) it isseen that olf * 9 ) Colp ) and from Lemma 12 ( c ) and the equation of = tf it

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

Page 1708

Since se is in H4m ) , it

Tito ) F = že . However , since feq is in Him + o - 1 ) , and since by ( 5 ) , ( Tito ) feq

= ge , it

...

Since se is in H4m ) , it

**follows**that there exists some F in H ( m + p ) such that (Tito ) F = že . However , since feq is in Him + o - 1 ) , and since by ( 5 ) , ( Tito ) feq

= ge , it

**follows**that fell = F is in H ( m + P ) ( C ) so that a fortiori , feq is in Am + P...

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 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 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 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