## 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 ( ) = lim x *

R ( ; T ' ) x = 0 , $ 0 Š - 00 it is

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 ( ) = lim x *

R ( ; T ' ) x = 0 , $ 0 Š - 00 it is

**seen**that x * x ( E ) = 0 for all & and all r * e ** .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

... uniform operator topology , and for which ( ii ) F ( F ) T = TF ( $ ) , ξε σ ( T ' ) . It is

range of E , that is , F ( $ ) E ( 0 ) = E ( 0 ) F ( F ) , ξερ ( Τ ) , for every Borel set o .

... uniform operator topology , and for which ( ii ) F ( F ) T = TF ( $ ) , ξε σ ( T ' ) . It is

**seen**from Corollary XV.3.7 that F ( x ) also commutes with the projections in therange of E , that is , F ( $ ) E ( 0 ) = E ( 0 ) F ( F ) , ξερ ( Τ ) , for every Borel set o .

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### Contents

SPECTRAL OPERATORS 1937 1941 1945 XV Spectral Operators | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

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