## 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

Since the cardinals are well ordered , there exists a projection G SE with m ( G ) =

no . Clearly G has uniform multiplicity no . Now from Zorn ' s lemma we obtain a

maximal family F of

Since the cardinals are well ordered , there exists a projection G SE with m ( G ) =

no . Clearly G has uniform multiplicity no . Now from Zorn ' s lemma we obtain a

maximal family F of

**disjoint**elements of B each having uniform multiplicity .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 | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 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 function 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