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

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Page 2300

Then , since the collection of finite sums of projections E ( n ; T ' ) is

bounded , it is clear from [ * ] that the collection of finite sums of projections E ( Mn

; T + P ) , n 2 K , is

Then , since the collection of finite sums of projections E ( n ; T ' ) is

**uniformly**bounded , it is clear from [ * ] that the collection of finite sums of projections E ( Mn

; T + P ) , n 2 K , is

**uniformly**bounded . Moreover , Erap ( Elan ; T ) – Elun ; T + P ...Page 2361

Thus , as in the third paragraph of the proof of Theorem 6 , it is sufficient to show

that F * ( a ) = ( ( T + P ) * — XI ) - 18 * is

P ) * ) . However , in the course of the proof of Theorem 6 it was established that ...

Thus , as in the third paragraph of the proof of Theorem 6 , it is sufficient to show

that F * ( a ) = ( ( T + P ) * — XI ) - 18 * is

**uniformly**bounded for each fe Sm ( ( T +P ) * ) . However , in the course of the proof of Theorem 6 it was established that ...

Page 2382

We shall now show that the second expression on the right of ( 14 ) converges to

zero as 14 →00 , ji remaining in P + ,

sequence of functions in Co [ 0 , 0 ) , each vanishing outside a bounded subset of

[ 0 ...

We shall now show that the second expression on the right of ( 14 ) converges to

zero as 14 →00 , ji remaining in P + ,

**uniformly**for 0 St < oo . Let { an } be asequence of functions in Co [ 0 , 0 ) , each vanishing outside a bounded subset of

[ 0 ...

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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