## 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 2169

uniform limit of analytic functions , it follows that this map is also a homomorphism

on the algebra of continuous functions . To see that it is a homomorphism on the

algebra of bounded

uniform limit of analytic functions , it follows that this map is also a homomorphism

on the algebra of continuous functions . To see that it is a homomorphism on the

algebra of bounded

**Borel**functions , note that for a fixed continuous function g ...Page 2188

It will be used frequently in what follows . 8 LEMMA . 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

It will be used frequently in what follows . 8 LEMMA . 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 ...Page 2262

+ Hence ( 5 ) holds if f is any bounded

völ . A repetition of this argument shows that ( 5 ) still holds if f and g are both

bounded

from ...

+ Hence ( 5 ) holds if f is any bounded

**Borel**function vanishing at = 0 and at d =völ . A repetition of this argument shows that ( 5 ) still holds if f and g are both

bounded

**Borel**functions vanishing at i = 0 and at do = völ . Since it is obviousfrom ...

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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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### Common terms and phrases

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