## Linear operators: Spectral operators |

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Page 2174

boundedness implied that S + T was a spectral operator, we would have a

contradiction to McCarthy's [2, 1] modification of Kakutani's example. In the

positive ...

**Hence**we have |ews+r>|=|e«se«T| ^ |e"s| |e"r| ^ MXM2 for all t e R.**Hence**if thisboundedness implied that S + T was a spectral operator, we would have a

contradiction to McCarthy's [2, 1] modification of Kakutani's example. In the

positive ...

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From this, it follows immediately that S — XI is closed, and

. 1.5, that (S — XI)'1 is closed. Since this latter operator is everywhere defined, it

follows by the closed graph theorem (II. 2.4) that it is bounded. Consequently, A ...

From this, it follows immediately that S — XI is closed, and

**hence**from Lemma XII. 1.5, that (S — XI)'1 is closed. Since this latter operator is everywhere defined, it

follows by the closed graph theorem (II. 2.4) that it is bounded. Consequently, A ...

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bounded operator which is compact if P(T — XI)~y is compact (cf. VI. 5.4). In all

cases (a), (b), (c) of the theorem, we may consequently pass from consideration

of the ...

**Hence**(P + N)(S - A7)"v = P(S - A/)"v + N{S-XI)'V = P(T-\I)-VL + N(S-\1)-* is abounded operator which is compact if P(T — XI)~y is compact (cf. VI. 5.4). In all

cases (a), (b), (c) of the theorem, we may consequently pass from consideration

of the ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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