## 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 1930

3 DEFINITION . A spectral measure E is said to be countably additive if for each 2

* in X * and each x in X the scalar set function x * E ( • ) x is countably additive on

the

3 DEFINITION . A spectral measure E is said to be countably additive if for each 2

* in X * and each x in X the scalar set function x * E ( • ) x is countably additive on

the

**domain**of E . + 4 COROLLARY . If the**domain**of a countably 1930 XV . 2 .Page 2087

( ii ) 68 Let As be the operator of the preceding exercise and let B be an arbitrary

bounded linear operator in Hp . Then ( i ) The operator As + B with

( 29 ) ( RN ) is the infinitesimal generator of a strongly continuous semi - group S

...

( ii ) 68 Let As be the operator of the preceding exercise and let B be an arbitrary

bounded linear operator in Hp . Then ( i ) The operator As + B with

**domain**( p ) T( 29 ) ( RN ) is the infinitesimal generator of a strongly continuous semi - group S

...

Page 2256

where C1 is a finite collection of closed Jordan curves bounding a

containing the union of o ( T ) and a neighborhood of infinity , C , being oriented

in the customary positive sense of complex variable theory . The curves C of the ...

where C1 is a finite collection of closed Jordan curves bounding a

**domain**Decontaining the union of o ( T ) and a neighborhood of infinity , C , being oriented

in the customary positive sense of complex variable theory . The curves C of the ...

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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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### Common terms and phrases

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