## 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

Restriction of a

mechanical three - body problem , XX . 6 ( 2501 ) , ( 2507 ) Quantum theory of

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

( 1938 ) ...

Restriction of a

**spectral operator**, spectrum of , XV . 8 . 5 ( 1956 ) Quantummechanical three - body problem , XX . 6 ( 2501 ) , ( 2507 ) Quantum theory of

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

( 1938 ) ...

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### Contents

SPECTRAL OPERATORS | 1924 |

An Operational Calculus for Bounded Spectral | 1941 |

Part | 1950 |

Copyright | |

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adjoint operator analytic apply arbitrary assume B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded operator Chapter clear closed commuting compact complex consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator discrete 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 Math Moreover multiplicity norm 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 uniform uniformly unique valued vector weakly zero