## Linear Operators: Spectral operators |

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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 ( V , Eq ) = Vam ( Ec ) for each family { E , } S D for which Va Eqe 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 ( V , Eq ) = Vam ( Ec ) for each family { E , } S D for which Va Eqe 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 ...

Page 2288

J . Dieudonné [ 20 ] had previously obtained a

where the adjoint X * of the underlying Banach space X is separable ( which

implies the separability of X ) . In connection with Lemma 3 . 18 we note that in

general X ...

J . Dieudonné [ 20 ] had previously obtained a

**multiplicity**theory in the casewhere the adjoint X * of the underlying Banach space X is separable ( which

implies the separability of X ) . In connection with Lemma 3 . 18 we note that in

general X ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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