## Linear Operators: Spectral operators |

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Page 2118

A generalized scalar

scalar

two ...

A generalized scalar

**operator**T e B ( X ) is said to be regular if it has a regular**spectral**distribution . Although it is not known whether or not every generalizedscalar

**operator**is regular ( unless the**spectrum**is sufficiently " thin " ) , given anytwo ...

Page 2249

Then , by Theorem 9 ( ii ) and Lemma 19 , T | E ( e ) X is a bounded

o ( T | E ( e ) X ) 2 Z . By the

Theorem VII . 3 . 11 ) and by Theorem 9 ( ii ) , Theorem 9 ( i ) , and Corollary 20 , it

...

Then , by Theorem 9 ( ii ) and Lemma 19 , T | E ( e ) X is a bounded

**operator**ando ( T | E ( e ) X ) 2 Z . By the

**spectral**mapping theorem for bounded**operators**(Theorem VII . 3 . 11 ) and by Theorem 9 ( ii ) , Theorem 9 ( i ) , and Corollary 20 , it

...

Page 2590

6 ( 2501 ) , ( 2507 ) Quasi - nilpotent operator , definition of , XV . 4 . 2 ( 1938 )

spectrum of , XV . 4 . 3 ( 1939 ) Quasi - nilpotent part of a

definition of , XV . 4 . 6 ( 1941 ) Radical part of a

XV . 4 .

6 ( 2501 ) , ( 2507 ) Quasi - nilpotent operator , definition of , XV . 4 . 2 ( 1938 )

spectrum of , XV . 4 . 3 ( 1939 ) Quasi - nilpotent part of a

**spectral operator**,definition of , XV . 4 . 6 ( 1941 ) Radical part of a

**spectral operator**, definition of ,XV . 4 .

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

31 other sections not shown

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adjoint operator analytic applications arbitrary assumed B-space Banach space belongs Boolean algebra Borel sets boundary bounded bounded operator Chapter clear closed commuting compact complex condition consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined domain elements equation equivalent established example exists extension fact finite follows formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal perturbation plane positive preceding present problem projections PROOF properties proved range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero