## 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 2264

**Multiplicity**Theory and Spectral Representation The methods and results of this section are due to Bade and are intimately dependent upon the ideas introduced in Section XVII.3 . A**multiplicity**theory for Boolean algebras of projections ...Page 2265

Let D be a dense ideal in B and m be a function on D to cardinals such that m ( 0 ) = 0 and m ( Va E. ) = Væm ( Ec ) for each family { Ex } $ D for which VaEge D. Then there is a unique

Let D be a dense ideal in B and m be a function on D to cardinals such that m ( 0 ) = 0 and m ( Va E. ) = Væm ( Ec ) for each family { Ex } $ D for which VaEge D. Then there is a unique

**multiplicity**function on B which is an extension ...Page 2283

Then a projection E in B has finite uniform

Then a projection E in B has finite uniform

**multiplicity**n if and only if its adjoint E * in B * has finite uniform**multiplicity**n . PROOF . It is sufficient to suppose E and E * satisfy the countable chain condition .### What people are saying - Write a review

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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