## 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 2209

The weakly

The weakly

**closed**operator algebra generated by a o - complete Boolean algebra B which satisfies the condition of the preceding lemma is the same as the uniformly**closed**algebra generated by B. PROOF . It is clear that every element in ...Page 2214

By Corollary VI.1.5 , the weakly

By Corollary VI.1.5 , the weakly

**closed**operator algebra W ( B ) generated by B is the same as the strongly**closed**algebra generated by B. Thus every A in W ( B ) is the strong limit of finite linear combinations of elements of B. It ...Page 2217

Let X ( x ) , X ( 2 ) be the

Let X ( x ) , X ( 2 ) be the

**closed**linear manifolds generated by the sets { Ex | E E B } , { Ex | Ee Bi } , respectively . Since B , is the strong closure of B , each element Ex with E in B , is contained in X ( x ) and thus X ( x ) 2X ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

31 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 contained continuous converges Corollary corresponding defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear 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