## 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

section are due to Bade and are intimately dependent upon the ideas introduced

in Section XVII . 3 . A

...

**Multiplicity**Theory and Spectral Representation The methods and results of thissection are due to Bade and are intimately dependent upon the ideas introduced

in Section XVII . 3 . A

**multiplicity**theory for Boolean algebras of projections in a B...

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 . ) = Vam ( Ed ) for each family { E } = D for which V . Ece 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 . ) = Vam ( Ed ) for each family { E } = D for which V . Ece D .

Then there is a unique

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

Then a projection E in B has finite uniform

in B * has finite uniform

* satisfy the countable chain condition . Also since each projection is the union ...

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 . Also since each projection is the union ...

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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### Common terms and phrases

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