## 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 2257

3 that E ( o ; T ) = E ( 70 ) ; R ) for each

clear that as o runs over the family K of all

runs over the family of all

3 that E ( o ; T ) = E ( 70 ) ; R ) for each

**compact**spectral set o of T . Moreover , it isclear that as o runs over the family K of all

**compact**open subsets of o ( T ) , 7 ( 0 )runs over the family of all

**compact**open subsets of o ( R ) which do not contain ...Page 2360

It will also be shown that T - v is

, it will follow that B ( u ) = R ( u ; T + P ) is

large , so that the theorem will be proved . Let u be in V ; . To show that IT ' ' R ( u

...

It will also be shown that T - v is

**compact**. From this , ( iii ) , and Theorem VI . 5 . 4, it will follow that B ( u ) = R ( u ; T + P ) is

**compact**for u in V , and i sufficientlylarge , so that the theorem will be proved . Let u be in V ; . To show that IT ' ' R ( u

...

Page 2462

The operator C is

is complete . Q . E . D . 12 LEMMA . If C is a

uniformly bounded sequence of operators in H converging strongly to zero ...

The operator C is

**compact**by Corollary V1 . 5 . 5 , and thus proof of Corollary 11is complete . Q . E . D . 12 LEMMA . If C is a

**compact**operator in H , and { Tn } is auniformly bounded sequence of operators in H converging strongly to zero ...

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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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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