## 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

Then since the

Then since the

**range**of T is closed , it follows from Corollary 12 that TX = X. By Lemma 3.1 the operator T is one - to - one . By Theorem II.2.2 , T has a bounded inverse and hence 0 € p ( T ) = p ( S ) and so SX = X. Next suppose that ...Page 1954

Q.E.D. a { 0 } It was shown in the course of the preceding proof that for an operator T with a closed

Q.E.D. a { 0 } It was shown in the course of the preceding proof that for an operator T with a closed

**range**the point = 0 is not in the spectrum of the operator V = T | E ( { 0 } ' ) X . Thus for all sufficiently small complex numbers 1 ...Page 2312

The closure of the

The closure of the

**range**of a densely defined linear operator T is the set of all x such that y * x = 0 whenever T * y * = 0 . Proof . If T * y * = 0 , then y * y = y * Tz = ( T * y * ) 2 = 0 for all y = Tz in the**range**of T , and hence ...### 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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