## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 1930

If Eis a Boolean algebra of subsets of the complex plane which contains the void

set and the whole plane , in short , if & is

a spectral measure E on & is called a resolution of the identity ( or a spectral ...

If Eis a Boolean algebra of subsets of the complex plane which contains the void

set and the whole plane , in short , if & is

**a field**of sets in the complex plane , thena spectral measure E on & is called a resolution of the identity ( or a spectral ...

Page 2145

First of all the spectral measure E is not necessarily a resolution of the identity for

T , for , even though it commutes with T , it may not satisfy the inclusion relation o

( To ) 58 ( cf . Definition XV . 2 . 2 , or Definition I below ) . Second ,

...

First of all the spectral measure E is not necessarily a resolution of the identity for

T , for , even though it commutes with T , it may not satisfy the inclusion relation o

( To ) 58 ( cf . Definition XV . 2 . 2 , or Definition I below ) . Second ,

**the field**S ( T )...

Page 2148

To complete the proof it will suffice to show that o ( To ) 5 5 for every Borel set 8 .

If € 8 , then , using ( D ) , the compact set do ( T ) may be covered by a set o in

To complete the proof it will suffice to show that o ( To ) 5 5 for every Borel set 8 .

If € 8 , then , using ( D ) , the compact set do ( T ) may be covered by a set o in

**the****field**S ( T ) with 1€ 7 . Since T is a spectral operator of class ( S ( T ) , X * ) , we ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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### Common terms and phrases

analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear clearly closure commuting compact complex consider constant contained converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established example exists extension fact finite follows formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum statement strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero