## 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 2162

Thus , in view of

that T has property ( D ) . According to Lemma 10 condition ( D ) will be satisfied if

the points regular relative to T are dense on To . Thus Lemmas 12 , 13 , 14 give ...

Thus , in view of

**Theorem**4.5 , to prove the present**theorem**it suffices to showthat T has property ( D ) . According to Lemma 10 condition ( D ) will be satisfied if

the points regular relative to T are dense on To . Thus Lemmas 12 , 13 , 14 give ...

Page 2403

concrete situation to be studied , to hypotheses ( a ) and ( c ) of

overall plan will be as follows . First we shall prove an inequality for integral

operators ( Lemma 5 below ) which is elementary in the sense that it relates only

to ...

concrete situation to be studied , to hypotheses ( a ) and ( c ) of

**Theorem**1 . Ouroverall plan will be as follows . First we shall prove an inequality for integral

operators ( Lemma 5 below ) which is elementary in the sense that it relates only

to ...

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exists almost everywhere for each fe Îy ( D , Y ) , and , using Lemma 5 once more

, the integral ( 74 ) 1,14,68 " , , 14:14 | 1524 ) de 15 ( 2 ) | dx ' dy dx dy ] exists for

almost all z " e D. Therefore , by Tonelli's

...

exists almost everywhere for each fe Îy ( D , Y ) , and , using Lemma 5 once more

, the integral ( 74 ) 1,14,68 " , , 14:14 | 1524 ) de 15 ( 2 ) | dx ' dy dx dy ] exists for

almost all z " e D. Therefore , by Tonelli's

**theorem**( III.11.14 ) , the integral ( ( 75 )...

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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