## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 2239

If x is in E ( 7 ) X as well as in E ( e ) X , it

bounded functions ( cf . ... ( fXē ) x , so that Qo is well defined on Ueeso E ( e ) X .

It thus

If x is in E ( 7 ) X as well as in E ( e ) X , it

**follows**from the operational calculus forbounded functions ( cf . ... ( fXē ) x , so that Qo is well defined on Ueeso E ( e ) X .

It thus

**follows**from Lemma 6 that T ( f ) is a closed , densely defined operator .Page 2246

it

domain of C . It is clear then that ( XI – C ) R ( a ) = x for x in H and R ( a ) ( 21 – C

) x = x for x in D ( C ) , so that R ( a ) = R ( a ; C ) and 1 € ( C ) . On the other hand ,

if X ...

it

**follows**that R ( 1 ) is a bounded operator whose range is contained in thedomain of C . It is clear then that ( XI – C ) R ( a ) = x for x in H and R ( a ) ( 21 – C

) x = x for x in D ( C ) , so that R ( a ) = R ( a ; C ) and 1 € ( C ) . On the other hand ,

if X ...

Page 2459

Statement ( b ) of our lemma

of our theorem

in Lac ( H ) , it would

Statement ( b ) of our lemma

**follows**at once . If xn Lac ( H ) ... Thus statement ( a )of our theorem

**follows**. If F is a bounded ... If ( il – H ) - 1 [ ac ( H ) were not densein Lac ( H ) , it would

**follow**by the Hahn - Banach theorem ( cf . Corollary II .### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 other sections not shown

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### Common terms and phrases

adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension fact finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero