## 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 2108

143 ] used the word “ spectral ” ) if there

measure u on X to A and a bounded Baire measurable function f : X TMC such

that a = S f ( x ) p ( dx ) . It follows that there

on a ...

143 ] used the word “ spectral ” ) if there

**exists**a compact space X , a spectralmeasure u on X to A and a bounded Baire measurable function f : X TMC such

that a = S f ( x ) p ( dx ) . It follows that there

**exists**a spectral measure va definedon a ...

Page 2405

... eldr ) . s ' s S S If he L , ( S , £ , y ) and 2 Sr < oo , it follows that the integral ( 14 )

( Ah ) ( 8 ) = S 14 ( 8 , 0 ) | ( 0 ) u ( dt )

, for the norm of an element f of L , ( S , E , p ) , we have | Ah , S { A } , \ h \ , .

... eldr ) . s ' s S S If he L , ( S , £ , y ) and 2 Sr < oo , it follows that the integral ( 14 )

( Ah ) ( 8 ) = S 14 ( 8 , 0 ) | ( 0 ) u ( dt )

**exists**for u - almost all s , and that , writing fl, for the norm of an element f of L , ( S , E , p ) , we have | Ah , S { A } , \ h \ , .

Page 2418

, the integral ( 74 ) 514 , 62 " , 2 ) { S , 43120 | 15 ( 2 ) | dx dy ] dx dy DOD

for almost all z " e D . Therefore , by Tonelli ' s theorem ( III . 11 . 14 ) , the integral

...

**exists**almost everywhere for each fe Īp ( D , Y ) , and , using Lemma 5 once more, the integral ( 74 ) 514 , 62 " , 2 ) { S , 43120 | 15 ( 2 ) | dx dy ] dx dy DOD

**exists**for almost all z " e D . Therefore , by Tonelli ' s theorem ( III . 11 . 14 ) , the integral

...

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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