## Linear Operators: Spectral operators |

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Page 1934

PROOF . Using Theorem 2 , we see that if o ( x ) is void , then x ( $ ) is

everywhere defined , single valued , and hence entire . Since , by VII . 3 . 4 , lim x

* x ( K ) = lim x * R ( E ; T ) x = 0 , $ + 00 it is

x * € X * .

PROOF . Using Theorem 2 , we see that if o ( x ) is void , then x ( $ ) is

everywhere defined , single valued , and hence entire . Since , by VII . 3 . 4 , lim x

* x ( K ) = lim x * R ( E ; T ) x = 0 , $ + 00 it is

**seen**that x * x ( 5 ) = 0 for all & and allx * € X * .

Page 2093

It is readily

even when F is closed ; an elementary counterexample is given in Colojoară and

Foiaş [ 4 ; p . 25 ) . If T is a spectral operator with resolution of the identity E and if

8 ...

It is readily

**seen**that X ( F ) is a linear manifold in X , but it need not be closedeven when F is closed ; an elementary counterexample is given in Colojoară and

Foiaş [ 4 ; p . 25 ) . If T is a spectral operator with resolution of the identity E and if

8 ...

Page 2163

8 , it is

F { E } E ( 0 , 01 ) SK sup F ( $ i ) – F ( $ i ) ) , where the supremum is taken over

those i and j for which 0 , 0 , is not void . If by the norm 71 is understood the ...

8 , it is

**seen**that , for some constant K , FLEJE ( ) – Ë FLEJE / 0 ) | = È Ž { F ( E . ) –F { E } E ( 0 , 01 ) SK sup F ( $ i ) – F ( $ i ) ) , where the supremum is taken over

those i and j for which 0 , 0 , is not void . If by the norm 71 is understood the ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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