## 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 2149

Condition ( A ) of Section 2 is satisfied if the spectrum of T is nowhere

the complex plane . PROOF . If the resolvent set is

or even continuous , extensions of R ( ; T ) x must coincide on their common ...

Condition ( A ) of Section 2 is satisfied if the spectrum of T is nowhere

**dense**inthe complex plane . PROOF . If the resolvent set is

**dense**, then any two analytic ,or even continuous , extensions of R ( ; T ) x must coincide on their common ...

Page 2156

are both

9,1 – T ) ^ x = 0 } is

( 1,1 - T ) Nx = 0 for i = 1 or i = 2 , so that to prove the present lemma it will be ...

are both

**dense**in X. Since M , is**dense**in X , the manifold ( ^ , I – T ) NM , + { x | (9,1 – T ) ^ x = 0 } is

**dense**in X , so ... 0 } is also**dense**in X. By Lemma 7 , ( x ) ay if( 1,1 - T ) Nx = 0 for i = 1 or i = 2 , so that to prove the present lemma it will be ...

Page 2159

The union of all intervals of constancy relative to T is an open set

PROOF . It is clear that the union of intervals of constancy is open . To see that it

is

εγ ...

The union of all intervals of constancy relative to T is an open set

**dense**in I ..PROOF . It is clear that the union of intervals of constancy is open . To see that it

is

**dense**, let y be a closed subarc of I , having positive length and let γη = { λο λοεγ ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

28 other sections not shown

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### Common terms and phrases

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 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