## 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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Results 1-3 of 89

Page 1953

If E ( { } ) = 0 , then ( AI – T ' ) X is dense in X.

is not dense then , by Corollary II.3.13 , there is an x * in X * with x * # 0 and ** TX

= 0 . Let x1 # 0 and define the operator A by the equation Ax = x * ( x ) x1 , so ...

If E ( { } ) = 0 , then ( AI – T ' ) X is dense in X.

**PROOF**. First suppose that = 0.If TXis not dense then , by Corollary II.3.13 , there is an x * in X * with x * # 0 and ** TX

= 0 . Let x1 # 0 and define the operator A by the equation Ax = x * ( x ) x1 , so ...

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

which it is used . 1 LEMMA ( A ) . If a , ß are complex numbers and x , y are

vectors in ...

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**ofwhich it is used . 1 LEMMA ( A ) . If a , ß are complex numbers and x , y are

vectors in ...

Page 2192

Let X be a weakly complete B - space . Let A ( T ) be the full algebra generated by

a family 7 of commuting spectral operators . If the Boolean algebra B ...

**PROOF**. This follows from Corollary 12 and Lemma 1 . Q.E.D. 14 COROLLARY .Let X be a weakly complete B - space . Let A ( T ) be the full algebra generated by

a family 7 of commuting spectral operators . If the Boolean algebra B ...

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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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