## Linear operators: Spectral operators |

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Results 1-3 of 87

Page 1953

})A. Thus A=0. Q.E.D. 12 Corollary. // E({X}) = 0, then {XI - T)X is dense in X.

First suppose that A = 0. If TX is not dense then, by Corollary II.3.13, there is an ...

**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. // E({X}) = 0, then {XI - T)X is dense in X.

**Proof**.First suppose that A = 0. If TX is not dense then, by Corollary II.3.13, there is an ...

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

it is used. 1 Lemma (A). If «, /? are complex numbers and x, y are vectors in X, ...

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 whichit is used. 1 Lemma (A). If «, /? are complex numbers and x, y are vectors in X, ...

Page 2395

... in C[0, oo) which vanish outside a bounded set. Then (R(X; T)f, g) has a

continuous extension to a neighborhood of the point A = 0 with the point A = 0

itself deleted, and this extension satisfies the estimate \(R(\;T)f,g)\=0{\\\-1) as |A|->-

0.

... in C[0, oo) which vanish outside a bounded set. Then (R(X; T)f, g) has a

continuous extension to a neighborhood of the point A = 0 with the point A = 0

itself deleted, and this extension satisfies the estimate \(R(\;T)f,g)\=0{\\\-1) as |A|->-

0.

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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