## 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 2236

Then , since T | E ( en ) X is bounded ,

calculus of bounded operators ( cf . VII . 3 . ... In the first case described in

( T ) ...

Then , since T | E ( en ) X is bounded ,

**statements**( i ) and ( ii ) and the functionalcalculus of bounded operators ( cf . VII . 3 . ... In the first case described in

**statement**( v ) , o ( T ) is a compact subset of U , and by Corollary XV . 3 . 5 , E ( o( T ) ...

Page 2239

# 0 , and h ( s ) = 0 otherwise . Then sh ( s ) 5 1 , so that by ( c ) the operator T ...

**Statement**( a ) clearly will follow from**statement**( b ) .**Statement**( b ) follows from**statement**( f ) ; indeed , let | f ( s ) 2 \ g ( s ) ] , and puth ( s ) = g ( s ) / S ( 8 ) if f ( 8 )# 0 , and h ( s ) = 0 otherwise . Then sh ( s ) 5 1 , so that by ( c ) the operator T ...

Page 2476

We therefore see that , as asserted ,

that gi ( a ) = 0 for all a key . Since ( H + V , H ) is a closed subspace of H ' ( cf .

Lemma 2 ) it follows that every s e H ' such that gi ( a ) = 0 for all a € e , belongs to

...

We therefore see that , as asserted ,

**statement**( 76 ) holds for each se H ' suchthat gi ( a ) = 0 for all a key . Since ( H + V , H ) is a closed subspace of H ' ( cf .

Lemma 2 ) it follows that every s e H ' such that gi ( a ) = 0 for all a € e , belongs to

...

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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