## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 2098

The following

Smart [ 2 ] . Let Te B ( x ) with o ( T ) = 0 . ( T ) = [ 0 , 1 ] with a bounded commuting

strongly continuous family E ( t ) , te [ 0 , 1 ] , of projections such that ( i ) E ( 0 ) ...

The following

**result**was proved by Sine [ 1 ] , using techniques similar to those inSmart [ 2 ] . Let Te B ( x ) with o ( T ) = 0 . ( T ) = [ 0 , 1 ] with a bounded commuting

strongly continuous family E ( t ) , te [ 0 , 1 ] , of projections such that ( i ) E ( 0 ) ...

Page 2214

Q . E . D . The preceding material of the present section was largely preliminary to

the following basic

THEOREM . Let B be a bounded Boolean algebra of projections in a weakly ...

Q . E . D . The preceding material of the present section was largely preliminary to

the following basic

**result**of Bade , which was described in the introduction . 18THEOREM . Let B be a bounded Boolean algebra of projections in a weakly ...

Page 2348

Then the present

Theorem 8 , Theorem 13 , and Lemma 15 once we showed that ( using the

notations of the hypothesis of Theorem 2 . 7 ) = 1 dona ( 11 . m . + dm ) 2 ( n − 2 )

/ n < 0o .

Then the present

**result**would follow from Theorem 2 . 7 , Corollary 2 . 8 ,Theorem 8 , Theorem 13 , and Lemma 15 once we showed that ( using the

notations of the hypothesis of Theorem 2 . 7 ) = 1 dona ( 11 . m . + dm ) 2 ( n − 2 )

/ n < 0o .

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

28 other sections not shown

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adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension fact finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero