## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 1951

Then every projection E ( 0 ) with 0 € ö belongs to J . If J is

also belong to J . PROOF . Let 0 € ō and let To = TE ( 0 ) | E ( o ) X , the restriction

of T to the invariant subspace E ( 0 ) X . Since o ( T . ) sē , it follows that 0 € p ( T ...

Then every projection E ( 0 ) with 0 € ö belongs to J . If J is

**closed**, then S and Nalso belong to J . PROOF . Let 0 € ō and let To = TE ( 0 ) | E ( o ) X , the restriction

of T to the invariant subspace E ( 0 ) X . Since o ( T . ) sē , it follows that 0 € p ( T ...

Page 2209

The weakly

algebra B which satisfies the condition of the preceding lemma is the same as the

uniformly

the ...

The weakly

**closed**operator algebra generated by a o - complete Booleanalgebra B which satisfies the condition of the preceding lemma is the same as the

uniformly

**closed**algebra generated by B . Proof . It is clear that every element inthe ...

Page 2217

Let X ( x ) , Xy ( 2 ) be the

} , { Ex Ee Bl } , respectively . Since B , is the strong closure of B , each element Ex

with E in B , is contained in X ( x ) and thus X ( x ) 2 X ( 2 ) . Evidently X ( x ) = x ...

Let X ( x ) , Xy ( 2 ) be the

**closed**linear manifolds generated by the sets { EX Ee B} , { Ex Ee Bl } , respectively . Since B , is the strong closure of B , each element Ex

with E in B , is contained in X ( x ) and thus X ( x ) 2 X ( 2 ) . Evidently X ( x ) = x ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

28 other sections not shown

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