## Linear Operators: Spectral operators |

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Page 2391

Thus , if de R , 191 > ? , and A ( ) # 0 , then ( XI – T ) - 1 = R ( 1 ; T ) exists and

equals R ( a ) , completing the proof of the present

COROLLARY . Let the hypotheses of

Suppose in the ...

Thus , if de R , 191 > ? , and A ( ) # 0 , then ( XI – T ) - 1 = R ( 1 ; T ) exists and

equals R ( a ) , completing the proof of the present

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

**Lemma**4 be satisfied and let 0 < d < 0 .Suppose in the ...

Page 2396

JO It follows from this formula just as in the proof of

following formula ( 14 ) ) that lim fu ( t ) = 0 , uniformly for ost < oo . 14100 HEP +

Hence , by formula ( 24 ) of the proof of

JO It follows from this formula just as in the proof of

**Lemma**1 ( cf . the paragraphfollowing formula ( 14 ) ) that lim fu ( t ) = 0 , uniformly for ost < oo . 14100 HEP +

Hence , by formula ( 24 ) of the proof of

**Lemma**3 , ĝu ( t ) ~ e - th ; ( t ) = - ime ...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 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