## 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 | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

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