## Linear Operators: Spectral operators |

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Page 2239

T ' ( fxz ) E ( e ) x = T ( fxě ) x , so that Qo is well defined on Ueeso E ( e ) X . It thus

, statement ( g )

T ' ( fxz ) E ( e ) x = T ( fxě ) 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**from Corollary 7 . Statement ( d ) is obvious . Letting ...Page 2246

it

domain of C . It is clear then that ( ^ l - C ) R ( 2x = x for x in H and R ( 2 ) ( al – C )

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

= n ...

it

**follows**that R ( a ) is a bounded operator whose range is contained in thedomain of C . It is clear then that ( ^ l - C ) R ( 2x = x for x in H and R ( 2 ) ( al – C )

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

= n ...

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Statement ( b ) of our lemma

statement ( a ) of our theorem

dense in Lac ( H ) , it would

. 3 .

Statement ( b ) of our lemma

**follows**at once . If xn e Lac ( H ) and ... Thusstatement ( a ) of our theorem

**follows**. If F is a ... If ( il – H ) - Eac ( H ) were notdense in Lac ( H ) , it would

**follow**by the Hahn - Banach theorem ( cf . Corollary II. 3 .

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### Contents

SPECTRAL OPERATORS 1937 1941 1945 XV Spectral Operators | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

29 other sections not shown

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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 defined Definition denote dense determined differential operator discrete domain elements equation equivalent established exists extension fact finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear 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