## 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 2188

Let E be a spectral measure in the complex B - space X which is defined and

countably additive on a o - field of subsets of a

measurable function defined on the complex plane . Then 5,868 ( ) E ( d ) ) = S , 4

...

Let E be a spectral measure in the complex B - space X which is defined and

countably additive on a o - field of subsets of a

**set**1 and let g be a bounded**Borel**measurable function defined on the complex plane . Then 5,868 ( ) E ( d ) ) = S , 4

...

Page 2189

Hence for an arbitrary g in EB ( 1 , 3 ) , equation ( ii ) holds for every characteristic

function f of a set in E. But the set of f for which ( ii ) holds is ... Now Ey is defined

and countably additive on the field of

Hence for an arbitrary g in EB ( 1 , 3 ) , equation ( ii ) holds for every characteristic

function f of a set in E. But the set of f for which ( ii ) holds is ... Now Ey is defined

and countably additive on the field of

**Borel sets**and it commutes with S ( f ) .Page 2233

Let { en } be an arbitrary increasing sequence of bounded

closures contained in U , such that E ( UK - 1 ( n ) = 1 . The operator f ( T ) is

defined by the equations D ( f ( T ) ) = { x ) lim f ( T | Elen ) X ) E ( en ) x exists } , f (

T ) x = lim f ...

Let { en } be an arbitrary increasing sequence of bounded

**Borel sets**withclosures contained in U , such that E ( UK - 1 ( n ) = 1 . The operator f ( T ) is

defined by the equations D ( f ( T ) ) = { x ) lim f ( T | Elen ) X ) E ( en ) x exists } , f (

T ) x = lim f ...

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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