## 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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Results 1-3 of 32

Page 1974

The formula ( iii ) of Lemma 3 shows that ( ii )

. Let ( a ) sup e - ess sup \ E ( o ; ( s ) ) ] = K < 00 , DESSES and suppose that for ...

The formula ( iii ) of Lemma 3 shows that ( ii )

**implies**( iii ) . It is clear that ( iii )**implies**( i ) , and so to prove the lemma it will suffice to prove that ( i )**implies**( ii ). Let ( a ) sup e - ess sup \ E ( o ; ( s ) ) ] = K < 00 , DESSES and suppose that for ...

Page 2129

... if ( a ) x Sy

dx = dy for all x , y in V and de R , 120 . If V is an ordered vector space under S ,

then the set K = { x € V10 x } is called XV . 16 2129 NOTES AND REMARKS.

... if ( a ) x Sy

**implies**that x + z Sy + z for all x , y , z in V , and ( b ) x Sy**implies**thatdx = dy for all x , y in V and de R , 120 . If V is an ordered vector space under S ,

then the set K = { x € V10 x } is called XV . 16 2129 NOTES AND REMARKS.

Page 2218

... weak convergence

generalized sequence of projections in a o - complete Boolean algebra of

projections in a B - space converges weakly to a projection , then it converges

strongly . PROOF .

... weak convergence

**implies**strong convergence . 27 THEOREM . If ageneralized sequence of projections in a o - complete Boolean algebra of

projections in a B - space converges weakly to a projection , then it converges

strongly . PROOF .

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 other sections not shown

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### Common terms and phrases

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 function 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