## 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 2068

It is clear that the linear space Lo = Lin L .

C of all integrable functions which coincide almost everywhere with a continuous

function clearly

It is clear that the linear space Lo = Lin L .

**satisfies**( 11 ) . Also the space Lo = LinC of all integrable functions which coincide almost everywhere with a continuous

function clearly

**satisfies**( 11 ) . Example 11 . 12 shows that the space Lo = Lin ...Page 2112

An example is given to show that not every operator admits a duality theory of

type 1 ; however , if T

the interior of F2 , then T does admit such a duality theory . It is a distinctly non ...

An example is given to show that not every operator admits a duality theory of

type 1 ; however , if T

**satisfies**the condition : ( a ) N ( F1 , T ) S M ( F2 , T ) , if Fisthe interior of F2 , then T does admit such a duality theory . It is a distinctly non ...

Page 2399

It is readily verified that ô ( t )

the above asymptotic relationships . Hence , if we let oz ( t ) be the unique

solution TO , = 0 such that oz ( t ) = ôz ( t ) for a St < oo , our lemma is proved . Q .

E . D ...

It is readily verified that ô ( t )

**satisfies**TO2 = 0 , and also**satisfies**the second ofthe above asymptotic relationships . Hence , if we let oz ( t ) be the unique

solution TO , = 0 such that oz ( t ) = ôz ( t ) for a St < oo , our lemma is proved . Q .

E . D ...

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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