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

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Page 2215

Thus the conclusion

LEMMA . Let B be a bounded o - complete Boolean algebra of projections in a B -

space X . Suppose that , for some sequence { x ; } in X , X = sp { Ex Ee B , i 2 1 } .

Thus the conclusion

**follows from Lemma**14 and Corollary 15 . Q . E . D . 21LEMMA . Let B be a bounded o - complete Boolean algebra of projections in a B -

space X . Suppose that , for some sequence { x ; } in X , X = sp { Ex Ee B , i 2 1 } .

Page 2308

10 and by

remark preceding Definition XIII . 2 . 29 , that S is closed . From this , it

immediately that S – XI is closed , and hence from

- 1 is ...

10 and by

**Lemma**XII . 1 . 6 ( a ) , it**follows**from Definition XIII . 2 . 17 and theremark preceding Definition XIII . 2 . 29 , that S is closed . From this , it

**follows**immediately that S – XI is closed , and hence from

**Lemma**XII . 1 . 5 , that ( 8 – XI )- 1 is ...

Page 2479

regarded as a subspace of the larger space H ' of

plainly H , may be regarded as the ... Indeed , it

is sufficiently large , we have ( 83 ) ( 101 - H2 ) - 1 = ( 101 – H ) - ( I – V ( 10I – H )

) - 1 ...

regarded as a subspace of the larger space H ' of

**Lemma**15 , while equallyplainly H , may be regarded as the ... Indeed , it

**follows**by**Lemma**13 that , if I dois sufficiently large , we have ( 83 ) ( 101 - H2 ) - 1 = ( 101 – H ) - ( I – V ( 10I – H )

) - 1 ...

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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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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