## 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

10 ) that T ( fxe ) X = T ( fxe ) E ( ē ) x = T ( fXeě ) x = T ( fxë ) E ( e ) x = T ' ( fXē ) x ,

so that Qo is well defined on Ueeso E ( e ) X . It thus

( f ) is a closed , densely defined operator . Moreover , statement ( g )

10 ) that T ( fxe ) X = T ( fxe ) E ( ē ) x = T ( fXeě ) x = T ( fxë ) E ( e ) x = T ' ( 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 . Moreover , statement ( g )

**follows**...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 XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear clearly closure commuting compact complex consider constant contained converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established example exists extension fact finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum statement strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero