## 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 1951

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**Corollary**VI.5.5 the compact operators form a closed twosided ideal in B ( X ) and so the present**corollary**is immediate . Q.E.D. 4**COROLLARY**.Page 1952

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**COROLLARY**. If A , T = 0 ( respectively TA , = 0 ) , then A , S = A , N = A , E ( 0 ) = 0 if 0 € 7 ( respectively SA , = NA . = E ( ) A , = 0 if 087 ) .Page 2192

This shows that o ( $ ( f ) ) 2 f ( 0 ) 2 0 $ ( 8 ) E ( O ) = 1 and completes the proof of the lemma , Q.E.D. 12

This shows that o ( $ ( f ) ) 2 f ( 0 ) 2 0 $ ( 8 ) E ( O ) = 1 and completes the proof of the lemma , Q.E.D. 12

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

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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