## 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 | 1924 |

An Operational Calculus for Bounded Spectral | 1941 |

Part | 1950 |

Copyright | |

9 other sections not shown

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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