## Linear Operators: Spectral operators |

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Page 2391

Thus , if le R , 11 > e , and A ( a ) # 0 , then ( WI – T ) -1 = R ( A ; T ) exists and

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

COROLLARY . Let the hypotheses of

Suppose in the ...

Thus , if le R , 11 > e , and A ( a ) # 0 , then ( WI – T ) -1 = R ( A ; 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

Then , by equations ( 22 ) and ( 23 ) of that same proof , | : 40 ) 14-1 { $ * 1968 )

ds 160 ) do ) ' + - 2 1u ( s – t ) q ( 8 ) ds It follows from this formula just as in the

proof of

Then , by equations ( 22 ) and ( 23 ) of that same proof , | : 40 ) 14-1 { $ * 1968 )

ds 160 ) do ) ' + - 2 1u ( s – t ) q ( 8 ) ds It follows from this formula just as in the

proof of

**Lemma**1 ( cf. the paragraph following formula ( 14 ) ) that lim ( t ) = 0 ...Page 2479

regarded as a subspace of the larger space H ' of

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

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

...

regarded as a subspace of the larger space H ' of

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

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

...

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### Contents

SPECTRAL OPERATORS 1937 1941 1945 XV Spectral Operators | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

32 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