## 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 1953

**PROOF**. If either AT = 0 or TA = 0 then , by the theorem , either A = AE ( { 0 } ) or A = E ( { 0 } ) A . Thus A = 0 . Q.E.D. 12 COROLLARY . If E ( { a } ) = 0 , then ( 21 – T ' ) X is dense in X.**Proof**. First suppose that = 0 .Page 2137

Even though ( A ) is taken as a standing assumption throughout this section , it will be indicated parenthetically in the statement of each lemma in the

Even though ( A ) is taken as a standing assumption throughout this section , it will be indicated parenthetically in the statement of each lemma in the

**proof**of which it is used . 1 LEMMA ( A ) . If a , ß are complex numbers and x ...Page 2192

This shows that o ( S ( f ) ) 2f ( o ) 2 n f ( 8 ) E ( ) = 1 and completes the

This shows that o ( S ( f ) ) 2f ( o ) 2 n f ( 8 ) E ( ) = 1 and completes the

**proof**of the lemma , Q.E.D. 12 COROLLARY . Let A be an algebra of operators in a weakly complete B - space X. Suppose that A is topologically and ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 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 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