## 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 ( 2 ; T ) is compact for every 1 € ( 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 ( 2 ; T ) is compact for every 1 € ( T ) ; ( c ) every

do in o ( T ) is a pole of

**finite**order vido ) of the resolvent and if , for some positive...

Page 2469

Therefore , for each n 2 1 and each & > 0 , the operator Vin , e ) is very smooth

and

= d ( 8 ) so that Vin . ( no ) - V « / * < 8/2 , and put y ( ) = V 4,0 ) -V , it follows that V

...

Therefore , for each n 2 1 and each & > 0 , the operator Vin , e ) is very smooth

and

**finite**. If we now choose n = n ( e ) so that IV ( n ) — V ] * < 8/2 , then choose d= d ( 8 ) so that Vin . ( no ) - V « / * < 8/2 , and put y ( ) = V 4,0 ) -V , it follows that V

...

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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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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