## 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 2081

where the supremum is taken over all finite sets { E ; } which are mutually

in the sense that E , Ex = 0 , j # k . Show that there exists a constant K such that v (

E , X , * * ) S K | 2 | 12 * . [ Hint : Examine the proof of Lemma III . 1 . 5 . ] ...

where the supremum is taken over all finite sets { E ; } which are mutually

**disjoint**in the sense that E , Ex = 0 , j # k . Show that there exists a constant K such that v (

E , X , * * ) S K | 2 | 12 * . [ Hint : Examine the proof of Lemma III . 1 . 5 . ] ...

Page 2265

ORE 72 0 That the extension is unique follows easily from the distributivity . Q . E .

D . 3 THEOREM . Let m be a multiplicity function on a complete Boolean algebra

B of projections in a B - space X . Then there is a unique family { En } of

ORE 72 0 That the extension is unique follows easily from the distributivity . Q . E .

D . 3 THEOREM . Let m be a multiplicity function on a complete Boolean algebra

B of projections in a B - space X . Then there is a unique family { En } of

**disjoint**...Page 2267

Since EEC we may express E as the union of a sequence of

En each of which is the carrier of a vector xn with 120ml = 1 . Define xo = { n - 1 2

- nxn . Clearly Exo = xo . We assert E is the carrier of xo . For suppose that Fe B ...

Since EEC we may express E as the union of a sequence of

**disjoint**projectionsEn each of which is the carrier of a vector xn with 120ml = 1 . Define xo = { n - 1 2

- nxn . Clearly Exo = xo . We assert E is the carrier of xo . For suppose that Fe B ...

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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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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