## 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 2323

The labor of verifying

elementary observations . By equations ( 1 ) , ( 4 ) , and our preliminary

normalization of boundary conditions , the terms of order p in the determinant N (

u ) are ...

The labor of verifying

**Hypothesis**1 may be simplified by using the followingelementary observations . By equations ( 1 ) , ( 4 ) , and our preliminary

normalization of boundary conditions , the terms of order p in the determinant N (

u ) are ...

Page 2397

PROOF . The theorem will follow as soon as it is shown that the

Theorem XVIII . ...

7 ( iv ) . It therefore only remains to establish

.

PROOF . The theorem will follow as soon as it is shown that the

**hypotheses**ofTheorem XVIII . ...

**Hypothesis**( ii ) has been established and is given by Lemma7 ( iv ) . It therefore only remains to establish

**hypothesis**( iii ) of Theorem XVIII . 2.

Page 2401

... or , using

Now , by

...

... or , using

**hypothesis**( b ) , to ( 4 ) P ( B ) – 1 ( B ) ' ( A1 ) = P ( A1 ) . Using**hypothesis**( c ) , we may write this last equation as q ( B – 4 ( B , A1 ) ) = Q ( A1 ) .Now , by

**hypothesis**, the map B + 4 ( B , A1 ) of A → A has norm at most M2 ( M1...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 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 function 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