## 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 | 1924 |

An Operational Calculus for Bounded Spectral | 1941 |

Part | 1950 |

Copyright | |

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adjoint operator analytic apply arbitrary assume B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded operator Chapter clear closed commuting compact complex consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator discrete 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 Math Moreover multiplicity norm positive preceding present problem projections PROOF properties proved range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniform uniformly unique valued vector weakly zero