## Linear Operators: Spectral operators |

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Page 2209

This means that A = An = S ( A ) E ( A1 ) for sufficiently large integers n , from

which it clearly follows that A is in the

Q . E . D . 15 COROLLARY . The weakly closed operator algebra generated by

a o ...

This means that A = An = S ( A ) E ( A1 ) for sufficiently large integers n , from

which it clearly follows that A is in the

**uniformly**closed algebra generated by B .Q . E . D . 15 COROLLARY . The weakly closed operator algebra generated by

a o ...

Page 2300

P 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 Elun ;

T + P ) , n 2 K , is

P 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 Elun ;

T + P ) , n 2 K , is

**uniformly**bounded . Moreover , L = 8 ( Eldin ; T ) – Elum ; T + P ...Page 2382

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

zero as lu ] → 00 , je remaining in P + ,

sequence of functions in C " [ 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 lu ] → 00 , je remaining in P + ,

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

[ 0 ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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adjoint operator analytic applications arbitrary assumed B-space Banach space belongs Boolean algebra Borel sets boundary bounded bounded operator Chapter clear closed commuting compact complex condition consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined 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 perturbation plane positive preceding present problem projections PROOF properties proved range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero