## Linear Operators: Spectral operators |

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Page 2188

Let E be a spectral measure in the complex B - space X which is defined and countably additive on a o - field of subsets of a

Let E be a spectral measure in the complex B - space X which is defined and countably additive on a o - field of subsets of a

**set**1 and let g be a bounded**Borel**measurable function defined on the complex plane .Page 2189

For a fixed characteristic function f the set of g in EB ( 1 , 2 ) for which ( ii ) holds is linear ; and , in view of ( i ) , it is ... Now E , is defined and countably additive on the field of

For a fixed characteristic function f the set of g in EB ( 1 , 2 ) for which ( ii ) holds is linear ; and , in view of ( i ) , it is ... Now E , is defined and countably additive on the field of

**Borel sets**and it commutes with S ( f ) .Page 2233

The operator f ( T ) of Definition 8 is closed , linear , and independent of the particular sequence of

The operator f ( T ) of Definition 8 is closed , linear , and independent of the particular sequence of

**Borel sets**used to define it . ( i ) For each**Borel set**e and each x in Dif ( T ) ) , E ( e ) D ( f ( T ) ) Ş D ( f ( T ) ) and E ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS 1937 1941 1945 XV Spectral Operators | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1931 |

Copyright | |

30 other sections not shown

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adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero