## Linear operators: Spectral operators |

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Page 2397

The theorem will follow as soon as it is shown that the

XVIII. 2. 34 are ...

iv). It therefore only remains to establish

The theorem will follow as soon as it is shown that the

**hypotheses**of TheoremXVIII. 2. 34 are ...

**Hypothesis**(ii) has been established and is given by Lemma 7 (iv). It therefore only remains to establish

**hypothesis**(iii) of Theorem XVIII. 2. 34.Page 2401

Using

Now, by

M2)~l. Thus, by Lemma VII. 3.4, the equation (6) B-j(B,A1)=A1 has a solution B e

...

Using

**hypothesis**(c), we may write this last equation as (5) <p{B-ftB,A1))=9(A1).Now, by

**hypothesis**, the map B-*<p(B, Aħ) of 91 -> 91 has norm at most M2(M1 +M2)~l. Thus, by Lemma VII. 3.4, the equation (6) B-j(B,A1)=A1 has a solution B e

...

Page 2449

On the other hand, assuming the truth of our assertion for a given n, it follows

from (6) that tn + 1 ^ t^ + 6M2tf. Since, by

2<i and our assertion follows. It also follows from (5) and

that ...

On the other hand, assuming the truth of our assertion for a given n, it follows

from (6) that tn + 1 ^ t^ + 6M2tf. Since, by

**hypothesis**, 6M2t1 ^ 1, we have 'n + 1 ^2<i and our assertion follows. It also follows from (5) and

**hypotheses**(d) and (e)that ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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