## 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 ( a ; 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 ( a ; T ' ) x must coincide on their common ...

Page 2156

are both

9 , 1 – T ' ) Nx = 0 } is

T ) ^ x = 0 } + { x | ( 1 , 1 – T ) x = 0 } is also

are both

**dense**in X . Since M , is**dense**in X , the manifold ( 1 , 1 – T ) \ M , + { x | (9 , 1 – T ' ) Nx = 0 } is

**dense**in X , so that ( 9 , 1 – T ) * ( 121 – T ) NX + { x | ( 9 , 1 –T ) ^ x = 0 } + { x | ( 1 , 1 – T ) x = 0 } is also

**dense**in X . By Lemma 7 , 0 ( 2 ) cy ...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

ey ...

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

**dense**in lo .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 T , having positive length and let Yn = { 10 doey ...

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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### Common terms and phrases

analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear clearly closure commuting compact complex consider constant contained converges Corollary corresponding countably additive defined Definition denote dense determined differential operator 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 manifold Math Moreover multiplicity norm positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum statement strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero