## 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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... as in the preceding section , ( 8 ) A , = S Â ( 8 ) e ( ds ) , 0€ E If { 0m } is a

sequence of sets in E satisfying ( 3 ) , then ( 9 ) Ap = lim Â ( s ) e ( ds ) , PED ( A ) ,

mom by Lemma 1 , and so the operator A is a type of

... as in the preceding section , ( 8 ) A , = S Â ( 8 ) e ( ds ) , 0€ E If { 0m } is a

sequence of sets in E satisfying ( 3 ) , then ( 9 ) Ap = lim Â ( s ) e ( ds ) , PED ( A ) ,

mom by Lemma 1 , and so the operator A is a type of

**unbounded**convolution .Page 2227

CHAPTER XVIII

the course of Chapters XII , XIII , and XIV that in order to apply the spectral theory

of Hermitian operators to ordinary and partial differential operators it is first ...

CHAPTER XVIII

**Unbounded**Spectral Operators 1 . Introduction It was shown inthe course of Chapters XII , XIII , and XIV that in order to apply the spectral theory

of Hermitian operators to ordinary and partial differential operators it is first ...

Page 2228

of Borel subsets of the complex plane . Let T be a linear operator whose domain

and range are contained in a complex B - space X . Then T is said to be a ...

**Unbounded**Spectral Operators ( i ) ( ii ) 1 DEFINITION . Let B denote the o - fieldof Borel subsets of the complex plane . Let T be a linear operator whose domain

and range are contained in a complex B - space X . Then T is said to be a ...

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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