## 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 2391

This shows that f = R ( A ) ( AI – T ) f for each fe D ( T ) . Thus , if le R , Al > ? , and

A ( X ) + 0 , then ( XI – T ) - 1 = R ( 1 ; T ) exists and equals R ( A ) , completing the

proof of the present

This shows that f = R ( A ) ( AI – T ) f for each fe D ( T ) . Thus , if le R , Al > ? , and

A ( X ) + 0 , then ( XI – T ) - 1 = R ( 1 ; T ) exists and equals R ( A ) , completing the

proof of the present

**lemma**. Q . E . D . 5 COROLLARY . Let the hypotheses of ...Page 2396

It follows from this formula just as in the proof of

following formula ( 14 ) ) that lim 1fu ( t ) = 0 , uniformly for Ost < 0 . HEP + Hence ,

by formula ( 24 ) of the proof of

+ ...

It follows from this formula just as in the proof of

**Lemma**1 ( cf . the paragraphfollowing formula ( 14 ) ) that lim 1fu ( t ) = 0 , uniformly for Ost < 0 . HEP + Hence ,

by formula ( 24 ) of the proof of

**Lemma**3 , ĝu ( t ) ~ e - ttu ; Ô ( t ) = - ime - itufu ( t )+ ...

Page 2479

regarded as a subspace of the larger space H ' of

plainly H , may be regarded as the restriction to H , of the operator H of

15 ( cf . ( 33 ) – ( 36 ) above ) . Let Q be the projection of H ' onto its subspace Hı ,

and ...

regarded as a subspace of the larger space H ' of

**Lemma**15 , while equallyplainly H , may be regarded as the restriction to H , of the operator H of

**Lemma**15 ( cf . ( 33 ) – ( 36 ) above ) . Let Q be the projection of H ' onto its subspace Hı ,

and ...

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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