## 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 2283

Then a projection E in B has

in B * has

* 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 2292

If T is discrete , then ( a ) its spectrum is a denumerable set of points with no

limit point ; ( b ) the resolvent R ( A ; T ) is compact for every 1 € 0 ( T ) ; ( c ) every

do in o ( T ) is a pole of

If T is discrete , then ( a ) its spectrum is a denumerable set of points with no

**finite**limit point ; ( b ) the resolvent R ( A ; T ) is compact for every 1 € 0 ( T ) ; ( c ) every

do in o ( T ) is a pole of

**finite**order vldo ) of the resolvent and if , for some ...Page 2469

Thus the proof of our lemma is complete . Q . E . D . Next we prove an important

inequality . 17 LEMMA . Suppose that we call on element f eH ' very smooth and

we ...

Thus the proof of our lemma is complete . Q . E . D . Next we prove an important

inequality . 17 LEMMA . Suppose that we call on element f eH ' very smooth and

**finite**if ( i ) there exists an integer n such that fi ( a ) = 0 for i > n ; ( ii ) for 1 şişn ,we ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

28 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 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