## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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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. ) = 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 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 | 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 defined Definition denote dense determined differential operator discrete 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