## 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 2299

Hence we find that for n

u ; T + P ) = B ( u ) . Since B ( u ) is clearly the product of the compact operator R (

u ; T ) and a bounded operator , it follows that T + P is a discrete operator .

Hence we find that for n

**sufficiently**large , each u in Cn is in plT + P ) and that R (u ; T + P ) = B ( u ) . Since B ( u ) is clearly the product of the compact operator R (

u ; T ) and a bounded operator , it follows that T + P is a discrete operator .

Page 2349

Hence , for m

+1 ] ) . kom Since , by ( * ) , Ιμη - μη +1 | 2πη ( 2πm ) : -1 , it is clear that mink + m

lien - Mal is bounded below by a function of the form Km " -1 for

...

Hence , for m

**sufficiently**large , min ( um - Mel 2 max ( Ium - Mm - 11 , I'm — Mom+1 ] ) . kom Since , by ( * ) , Ιμη - μη +1 | 2πη ( 2πm ) : -1 , it is clear that mink + m

lien - Mal is bounded below by a function of the form Km " -1 for

**sufficiently**large...

Page 2360

It will be shown below that \ T''R ( u ; T ) A | = } for p in V , and i

From this it will then follow as above that the function f ( u ) R ( u ; T + P ) f is

uniformly bounded . It will also be shown that T - v is compact . From this , ( iii ) ,

and ...

It will be shown below that \ T''R ( u ; T ) A | = } for p in V , and i

**sufficiently**large .From this it will then follow as above that the function f ( u ) R ( u ; T + P ) f is

uniformly bounded . It will also be shown that T - v is compact . From this , ( iii ) ,

and ...

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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