## 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 2148

If € 8 , then , using ( D ) , the compact set do ( T ) may be covered by a set o in the

field S ( T ) with 1€ 7 . Since T is a spectral operator of class ( S ( T ) , X * ) , we

have o ( T . ) Sõ and

...

If € 8 , then , using ( D ) , the compact set do ( T ) may be covered by a set o in the

field S ( T ) with 1€ 7 . Since T is a spectral operator of class ( S ( T ) , X * ) , we

have o ( T . ) Sõ and

**consequently**, is in p ( T . ) , which means that XI - T is a one...

Page 2323

terms of order p in the determinant N ( u ) become the same as the coefficients ap

, bp , Cp in the determinant Ñ ( u ) = a , etu + bpe - lu + cp of the matrix ( ÔNak ) ...

**Consequently**, by factoring out a factor ( iu ) ' , the coefficients ap , bp , Cp of theterms of order p in the determinant N ( u ) become the same as the coefficients ap

, bp , Cp in the determinant Ñ ( u ) = a , etu + bpe - lu + cp of the matrix ( ÔNak ) ...

Page 2343

minors of order v , we find that the expansion contains only two non - vanishing

terms . Thus our 2v X 2v determinant may be expressed as P2P , FQ1Q2 , where

...

**Consequently**, if we use Lagrange ' s rule to expand this 2v x 2v determinant byminors of order v , we find that the expansion contains only two non - vanishing

terms . Thus our 2v X 2v determinant may be expressed as P2P , FQ1Q2 , where

...

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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