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

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Page 1986

By considering the integral as an iterated integral , it is seen that it suffices to

du S - U sin at = lim - PIS – t ) - dt , _ and so equation ( 5 ) may be

By considering the integral as an iterated integral , it is seen that it suffices to

**establish**( 5 ) in the case N = 1 . ... sin als - - u ) 1 po = lim - 2 + 00 TT 1 - 0 - plu )du S - U sin at = lim - PIS – t ) - dt , _ and so equation ( 5 ) may be

**established**by.Page 2101

Such a formula has been

) the Boolean algebra generated by E , and E , is bounded , ( ii ) X is weakly

complete , and ( iii ) the E - measure of the boundary of o is zero . Further work

along ...

Such a formula has been

**established**by Foguel [ 1 ] under the hypotheses that ( i) the Boolean algebra generated by E , and E , is bounded , ( ii ) X is weakly

complete , and ( iii ) the E - measure of the boundary of o is zero . Further work

along ...

Page 2212

We next

fz Xocp ) so that , using ( vii ) , we have g ( 1 ) = 0 for 1€ 0 , 0 ( F ) = 0ft . Also T ( g )

... Thus if fz ( a ) = fw ( a ) on 0 , = ow , equation ( ix ) will be

We next

**establish**the equation ( viii ) fro = f : Xocp ) , Fe B . To prove this , let g =fz Xocp ) so that , using ( vii ) , we have g ( 1 ) = 0 for 1€ 0 , 0 ( F ) = 0ft . Also T ( g )

... Thus if fz ( a ) = fw ( a ) on 0 , = ow , equation ( ix ) will be

**established**. Hence ...### 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 formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm 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