## 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 2188

It will be used frequently in what follows . 8 LEMMA . 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

It will be used frequently in what follows . 8 LEMMA . 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 ...Page 2189

Now E , is defined and countably additive on the field of

commutes with S ( f ) . Thus to see that E , is the resolution of the identity for S ( f )

it will suffice to show that ( iv ) o ( S ( f ) | E2 ( 8 ) X ) 5 5 for every

complex ...

Now E , is defined and countably additive on the field of

**Borel sets**and itcommutes with S ( f ) . Thus to see that E , is the resolution of the identity for S ( f )

it will suffice to show that ( iv ) o ( S ( f ) | E2 ( 8 ) X ) 5 5 for every

**Borel set**8 ofcomplex ...

Page 2233

By virtue of the equation f ( T | E ( e ) X ) x = f ( T | E ( ē ) X ) , which has been

established for x in E ( e ) X n Esē ) X , we may define a single valued linear

operator Qo on Ue E ( e ) X , where e varies over the family of bounded

whose ...

By virtue of the equation f ( T | E ( e ) X ) x = f ( T | E ( ē ) X ) , which has been

established for x in E ( e ) X n Esē ) X , we may define a single valued linear

operator Qo on Ue E ( e ) X , where e varies over the family of bounded

**Borel sets**whose ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 other sections not shown

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### Common terms and phrases

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 countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension fact finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit 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